Abstract theory extends optimal transport to Banach lattices.
problem Generalize optimal transport theory to Banach lattices.
method Abstract framework, duality theory, Banach lattice, order unit.
result Characterization of dual elements for generalized optimal transport.
Develops a new method for robust risk measurement by averaging nearby payoffs.
problem Measuring risk under uncertainty with a focus on robustness.
method Averaging nearby payoffs weighted by a chosen metric.
result The method leads to a convex risk measure and provides stability under large neighborhoods.
Abstract framework for no-arbitrage concepts in topological vector lattices.
problem Generalization of no-arbitrage concepts in topological vector lattices.
method Imposing a structural condition on trading strategies and deriving abstract FTAP.
result NUPBR, NAA1, and NA1 may not be equivalent in general setting. The paper finds the smallest order closed sublattice containing a given sublattice in vector lattices.
problem Identifying the smallest order closed sublattice containing a given sublattice in vector lattices.
method Analyzing the order closure and second order closure of sublattices.
result The smallest order closed sublattice containing a sublattice Y is often the second order closure of Y. We identify a large class of Orlicz spaces X for which the topology σ(X,Xn∼) fails the C-property introduced in [7]. We also establish a variant of the C-property and use it to prove a w∗-representation theorem for proper convex increasing functionals on dual Banach lattices that satisfy a suitable version …
Study Banach-Lie groupoids, revealing new insights.
problem Understanding infinite-dimensional manifolds.
method Adapt classical Lie groupoid results to Banach-Lie groupoids.
result Locally transitive Banach-Lie groupoids offer new perspectives.
Proves linear extension of isometries in smooth 2D Banach spaces.
problem Linear extension of isometries in absolutely smooth 2D Banach spaces.
method Analyzes isometries between unit spheres of smooth Banach spaces.
result Any isometry extends to a linear isometry of Banach spaces.
Regulated curves on Banach manifolds with continuous projections and regulated derivatives are studied.
problem Regulated curves on Banach manifolds with continuous projections and regulated derivatives.
method Building a Banach manifold structure on the set of such curves.
result Existence of a 'local addition' on such a manifold for any Banach manifold.
The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.
problem Defining projective analogues of Lie bialgebras and Poisson-Lie groups.
method Introducing projective tensor products and adapting classical notions to these structures.
result Every quasi-triangular projective r-matrix gives rise to a projective Banach Lie bialgebra.
Integrability criterion for projective limits of Banach distributions on Fréchet manifolds.
problem Integrability of projective limits of involutive bundles on Banach manifolds.
method An integrability criterion for a projective limit of Banach distributions.
result Result of integrability of projective limit of involutive bundles on a projective sequence of Banach manifolds.
Formalizes integral curves on Banach manifolds in Lean.
problem Existence and uniqueness of integral curves on Banach manifolds.
method Formalized differential equations on Banach spaces, then generalized to Banach manifolds.
result Established theorems for integral curves on Banach manifolds.
A new method for clearing liability networks using sheaves on directed hypergraphs.
problem Clearing in liability networks using a novel mathematical approach.
method Associate a liability sheaf on a directed hypergraph to a liability network, identifying clearing configurations as global sections of this sheaf.
result Clearing configurations are precisely the global sections of the sheaf, and the sheaf construction is functorial under change of coefficient category.
First, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.
The paper studies symplectic forms on projective limits of Banach bundles and their Darboux Theorem.
problem Conditions for weak symplectic forms on projective limits of Banach bundles.
method Analyzing projective sequences of Banach bundles and applying Darboux Theorem.
result Necessary and sufficient conditions for the Darboux Theorem on projective limits of Banach manifolds.
Constructs Poisson structure on Banach Lie algebroid predual.
problem Linear Poisson structure on Banach Lie algebroid predual.
method Alternative approach to existing sub-Poisson structure on dual bundle.
result Existence of queer Banach Lie algebroids.
Researchers develop neural networks for approximating functions in Banach spaces.
problem Approximating Banach space valued continuous functions.
method Quasi-interpolation Banach space valued neural network operators using algebraic sigmoid functions.
result Jackson type inequalities for function approximation.
Examines different approaches to Poisson structures in Banach spaces.
problem Exploring various definitions of Poisson structures in Banach spaces.
method Presenting and comparing different definitions of Poisson structures.
result Illustrates differences between existing definitions with examples.
This paper is devoted to the framework of direct limit of anchored Banach bundles over a convenient manifold which is a direct limit of Banach manifold. In particular we give a criterion of integrability for distributions on such convenient manifolds which are locally direct limits of particular sequences of Banach anc…
A {1}-structure on a Banach manifold M (with model space E) is an E-valued 1-form on M that induces on each tangent space an isomorphism onto E. Given a Banach principal bundle P with connected base space and a {1}-structure on P, we show that its automorphism group can be turned into a Banach-Lie group acting smoothly…
Banach fibrations and Nijenhuis operators studied for vanishing torsion.
problem Understanding Nijenhuis operators on Banach fibrations and their properties.
method Analyzing Nijenhuis operators on Banach fibrations and their projectability.
result Vanishing of Nijenhuis torsion on N0 implies vertical values for N. Develops risk measures on Lipschitz spaces for financial positions.
problem Lack of standard cash-additive methods in Lipschitz spaces.
method Proposes Lipschitz-free space, uses additivity along benchmark-deviation instruments.
result Derives dual representations for convex and coherent risk measures.
Differential structure on partial isometries over Grassmannian constructed.
problem No specific problem stated; abstract focuses on method and result.
method Construction of differential structure on partial isometries over restricted Grassmannian.
result Set of partial isometries over restricted Grassmannian becomes a Banach Lie groupoid.
Analytic functions on Banach spaces with Lojasiewicz exponent 1/2 are Morse-Bott.
problem Characterizing analytic functions on Banach spaces with specific gradient properties.
method Proof of converse to the Morse-Bott property for analytic functions on Banach spaces using Lojasiewicz gradient inequality and Morse Lemma.
result Analytic functions on Banach spaces with Lojasiewicz exponent 1/2 are Morse-Bott.
This paper concerns the problem of integrability of non closed distributions on Banach manifolds. We introduce the notion of weak distribution and we look for conditions under which these distributions admit weak integral submanifolds. We give some applications to Banach Lie algebroid and Banach Lie-Poisson manifold. T…
Study projective and direct limits of Banach structures with connections to G-structures.
problem Understanding connections between Banach structures and G-structures. method Endow projective and direct limits with Fréchet or convenient structures and study connections.
result Illustrated examples demonstrate the study of projective and direct limits.
A Banach symmetric space in the sense of O. Loos is a smooth Banach manifold M endowed with a multiplication map μ:M×M→M such that each left multiplication map μx:=μ(x,⋅) (with x∈M) is an involutive automorphism of (M,μ) with the isolated fixed point x. We show that morphisms of …
Extends Gaussian process theory to Banach spaces.
problem Extending Gaussian process theory to Banach spaces.
method Investigates the connection between Gaussian processes and Gaussian random elements in reproducing kernel Banach spaces.
result Characterizes positive definite functions that arise from covariance operators in Banach space setting.
New structure on unitary group of Hilbert space.
problem No specific problem stated; constructing a new structure.
method Constructing a Banach Poisson-Lie group structure.
result Banach Poisson-Lie group structure on unitary group of Hilbert space.
The study integrates Banach manifolds into H-manifolds, integrating Lie algebras into H-groups.
problem Integrating Banach manifolds and Lie algebras into H-manifolds and H-groups.
method Investigating quotients of Banach manifolds with free actions of pseudogroups of local diffeomorphisms.
result Every real Banach-Lie algebra can be integrated to an H-group.
Study sub-Riemannian structures on Banach manifolds, extending controllability and geodesic flow results.
problem Analyzing sub-Riemannian structures on Banach manifolds.
method Define sub-Riemannian structures, extend Chow-Rashevski theorem, and provide conditions for Hamiltonian geodesic flow.
result Extensions of the Chow-Rashevski theorem for exact controllability and conditions for Hamiltonian geodesic flow in infinite-dimensional setting.
Extends variational principle to tensor Banach spaces.
problem High-dimensional partial differential equations and minimization problems.
method Describes tensor product as disjoint connected components, each modeled as a Banach manifold.
result Extension of Dirac-Frenkel variational principle to topological tensor spaces.
The article studies effective Reifenberg theorems in Hilbert and Banach spaces.
problem Understanding measures in Hilbert and Banach spaces.
method Analyzes Reifenberg theorems for measures in Hilbert and Banach spaces, providing conditions for rectifiability.
result Conditions for rectifiability of measures in Banach spaces, including a power gain for uniformly smooth spaces.
Robust SVM optimization in Banach spaces tackles classification uncertainty.
problem Binary classification in Banach spaces with uncertainty.
method Generalization of SVM results to Banach spaces, Representer Theorem, strong duality, Nash equilibrium formulation.
result Generalization of SVM results to Banach spaces, including Representer Theorem and strong duality.
The paper uses Banach spaces to analyze neural networks.
problem Understanding the function spaces of neural networks.
method Theory of reproducing kernel Banach spaces.
result Representer theorem for wide class of Banach spaces.
We investigate some basic questions concerning the relationship between the restricted Grassmannian and the theory of Banach Lie-Poisson spaces. By using universal central extensions of Lie algebras, we find that the restricted Grassmannian is symplectomorphic to symplectic leaves in certain Banach Lie-Poisson spaces, …
Introduces generalized almost statistical convergence and its properties.
problem Developing a new convergence concept for sequences.
method Introducing generalized almost statistical convergence and proving its properties.
result Existence of a GAS convergent sequence that is neither statistical nor almost convergent.
Smooth maps between manifolds form a Banach manifold.
problem Understanding the structure of maps between manifolds.
method Detailed rigorous proof of the smooth manifold structure.
result The set of k times continuously differentiable maps between manifolds forms a smooth Banach manifold. New Poisson brackets defined for Banach manifolds that can use higher-order derivatives.
problem Constructing Poisson brackets with higher-order derivatives on Banach manifolds.
method Method to construct Poisson brackets on Banach manifolds with dependence on higher-order derivatives.
result Counterexamples to the Leibniz property implying the existence of a Poisson tensor.
We extend Banach's fixed point theorem to establish convergence of iterative methods under carefully chosen metrics.
problem Limited applicability of Banach's fixed point theorem in non-convex problems.
method Developed a strong converse theorem to Banach's fixed point theorem, showing its universal applicability for convergence of iterative methods.
result We show that, whenever an iterative map globally converges to a unique fixed point, there exists a metric under which the iterative map is contracting and can be used to bound the convergence rate.
We introduce a notion of p-rough integrator on any Banach manifolds, for any p≥1, which plays the role of weak geometric Holder p-rough paths in the usual Banach space setting. The awaited results on rough differential equations driven by such objects are proved, and a canonical representation is given if the man…
We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. As an application, we extend some known results concerning existen…
This paper extends mirror descent to Banach spaces with reproducing kernels.
problem Optimizing in Banach spaces with reproducing kernels.
method Mirror descent algorithm adapted for Banach spaces with reproducing kernels.
result Mirror descent achieves linear convergence in certain conditions and standard convergence in a constrained setting.
Groups with specific subgroup actions have proper actions on uniformly convex spaces.
problem Proper actions of hyperbolic relative groups on Banach spaces.
method Affine isometric actions on uniformly convex Banach spaces.
result Groups with proper actions on uniformly convex spaces.
The study improves error bounds for statistical learning methods using weakly dependent sums.
problem Investigating error bounds for spectral regularization methods.
method Obtained a Bernstein-type inequality for Banach-valued sums under weak dependence and smoothness assumptions.
result Improved error upper bounds for spectral regularization methods trained on τ−mixing processes. Generalized Stacey-Roberts lemma for Banach manifolds.
problem Constructing Lie groupoids of smooth mappings in infinite-dimensional geometry.
method Generalization of the Stacey-Roberts lemma to Banach manifolds with smooth partitions of unity.
result Remedied an error in the original proof for finite-dimensional setting.
Paper solves support vector regression in Banach spaces using tensor-kernels.
problem Support vector regression in Banach function spaces.
method Fenchel-Rockafellar duality theory and tensor-kernel representation.
result Explicit formulation of dual problem and optimality conditions.
Study spectral and index properties of Hodge-Dirac operator on compact manifolds.
problem Investigate spectral and index-theoretic properties of Hodge-Dirac operator on compact Riemannian manifolds.
method Establish bisectoriality and H∞ functional calculus without curvature assumptions. result Prove compact Banach spectral triple and recover classical topological invariants as Lp-indices. Deep neural networks define suitable reproducing kernel Banach spaces.
problem Characterizing the function spaces of deep neural networks.
method Reproducing kernel Banach spaces and variational results.
result Deep neural networks define suitable reproducing kernel Banach spaces.