We prove implicit function theorems for mappings on topological vector spaces over valued fields. In the real and complex cases, we obtain implicit function theorems for mappings from arbitrary (not necessarily locally convex) topological vector spaces to Banach spaces.
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,695 papers · 148 categories
Trend · papers per month
48 results for “topological vector spaces”
Identifies submanifolds as topological spheres in hyperbolic space.
problem Characterizing submanifolds as topological spheres in hyperbolic space.
method Defines conditions on Ricci curvature and mean curvature vector length.
result Identifies submanifolds as topological spheres under given conditions.
Study geometric flows with varying parameters and prove continuous dependence.
problem Continuous dependence of flows on parameters in geometric settings.
method Derived suitable topologies for vector fields and flows, proved new continuous dependence.
result Proved continuous dependence of flows on parameters in a general topological space.
Differential topology of the spaces of asymptotically stable vector fields and Lyapunov functionsmath.DS
Study the topology of stable vector fields and Lyapunov functions on R^n.
problem Topology of stable vector fields and Lyapunov functions on R^n.
method Differential topology, Lyapunov theory, and results on diffeomorphism groups of discs.
result Path-connected and simply connected spaces of stable vector fields for n≠4,5 and weakly contractible for n≤3.
Universal approximation theorem for neural networks with inputs from a topological vector spacecs.LG
TVS-FNNs can approximate any continuous function on expanded input spaces.
problem Processing a broader range of inputs like sequences and matrices.
method Proving a universal approximation theorem for TVS-FNNs.
result TVS-FNNs can approximate any continuous function on expanded input spaces.
Study on topological rigidity of ALE vector bundles with specific conditions.
problem Classifying ALE vector bundles with asymptotically conical total spaces.
method Topological classification and geometric analysis of ALE vector bundles.
result Only 2-sphere, projective plane, and open contractible manifolds admit ALE tangent bundles.
Study the complexity of horizontality in 4-torus vector bundles.
problem Classify topological holonomy groups in SO(3).
method Analyze twistor spaces and oriented vector bundles over 2-torus.
result Discover many topological holonomy groups in SO(3) with noncommutative pairs.
FibeRed reduces complex data dimensions while preserving topology.
problem Hard embedding of topologically complex datasets in low-dimensional Euclidean space.
method Modeling datasets with vector bundles, reducing fibers while preserving topology.
result FibeRed learns topologically faithful embeddings in lower dimensions than existing methods.
Space of hyperbolic surfaces is path-connected.
problem Topology of hyperbolic surfaces and their subspaces.
method Constructing paths using Fenchel-Nielsen coordinates and shrinking curves.
result Path-connectivity of the space of hyperbolic surfaces.
SOM-VQ tokenizes discrete models with semantic structure and navigable topology.
problem Lack of semantic structure in vector quantized representations limits interpretable human control.
method Combines vector quantization with Self-Organizing Maps to learn discrete codebooks with explicit topology.
result SOM-VQ produces more learnable token sequences and provides an explicit navigable geometry in code space.
Novel metric space magnitude and weighting vectors improve machine learning tasks.
problem Improving machine learning algorithms using novel metric space concepts.
method Metric space magnitude and weighting vectors for better machine learning.
result The weighting vector effectively detects boundaries and improves classic machine learning tasks.
Develops a new exponential map for time-varying vector fields.
problem Lack of global flows for general time-varying vector fields.
method Categorical development of spaces of vector fields and flows, allowing for systematic localisation.
result Derives the homeomorphism of the exponential map for vector fields with measurable time-dependence.
Lyapunov 1-forms on orbifolds help understand flows on compact spaces.
problem Understanding flows on orbifolds using Lyapunov 1-forms.
method Introducing Lyapunov 1-forms, using asymptotic cycles and chain-recurrent sets.
result Existence of a Lyapunov 1-form in a prescribed cohomology class for compact orbifolds.
Linear Microbundlesmath.DG
Extends calculus to topological manifolds using generalized functions.
problem Proving the existence of non-singular generalized tangent vector fields on spheres.
method Develops a theory of generalized functions and applies it to continuous maps between topological spaces.
result Shows coherence between non-existence of smooth vector fields on spheres and existence of generalized ones.
Topological Data Analysis in Text Classification: Extracting Features with Additive Informationcs.LG
Paper explores applying TDA to text classification, improving model performance.
problem Applying TDA to text classification is challenging due to the complexity of text geometry.
method Used word embeddings and TF-IDF vectors to extract topological features from text.
result Topological features improve classification results, especially in ensemble models.
Proves every equivariant vector bundle over toric manifolds is a Klyachko bundle.
problem Characterizing equivariant vector bundles over toric manifolds.
method Analyzes topological and smooth equivariant vector bundles over toric manifolds.
result Every equivariant vector bundle is a Klyachko bundle.
In this paper we give a characterization of 2-dimensional topological field theories over a space as Frobenius bundles with connections over , the free loop space of . This is a generalization of the folk theorem stating that 2-dimensional topological field theories (over a point) are described by finite-dim…
Paper introduces stratified vector bundles and their properties.
problem Understanding singular spaces and their vector bundles.
method Characterization via monoid actions and examples from various fields.
result Functorial properties extended to the stratified case.
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, NAA, and NA may not be equivalent in general setting.
If a characteristic class for two vector bundles over the same base space does not coincide, then the bundles are not isomorphic. We give under rather common assumptions a lower bound on the topological dimension of the set of all points in the base over which a morphism between such bundles is not bijective. Moreover,…
New method reduces clustering time and improves accuracy.
problem High time and space complexity in spectral clustering.
method Approximate spectral clustering using GNG network topology.
result Equal or better clustering performance than traditional SC.
Weighting vectors for machine learning: numerical harmonic analysis applied to boundary detectioncs.LG
New method uses weighting vectors for efficient boundary and outlier detection.
problem Boundary and outlier detection in machine learning.
method Recast metric space magnitude as weighting vector, solve kernelized SVM, apply nearest neighbor methods.
result Weighting vector can be efficiently approximated in linear time, outperforming state-of-the-art techniques.
PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signaturesstat.ML
Persistence diagrams, the most common descriptors of Topological Data Analysis, encode topological properties of data and have already proved pivotal in many different applications of data science. However, since the (metric) space of persistence diagrams is not Hilbert, they end up being difficult inputs for most Mach…
Under certain topological assumptions, we show that two monotone Lagrangian submanifolds embedded in the standard symplectic vector space with the same monotonicity constant cannot link one another and that, individually, their smooth knot type is determined entirely by the homotopy theoretic data which classifies the …
We study the topology of moduli spaces of closed linkages in \R^d depending on a length vector \ell\in \R^n. In particular, we use equivariant Morse theory to obtain information on the homology groups of these spaces, which works best for odd d. In the case d=5 we calculate the Poincare polynomial in terms of combinato…
Topological Data Analysis (TDA) is a recent and growing branch of statistics devoted to the study of the shape of the data. In this work we investigate the predictive power of TDA in the context of supervised learning. Since topological summaries, most noticeably the Persistence Diagram, are typically defined in comple…
Paper introduces stable vectorization for multiparameter PH using signed barcodes.
problem Lack of stable vectorization methods for multiparameter persistent homology.
method Signed barcodes as measures for stable vectorization of MPH.
result Stable feature vectors from signed barcodes improve performance in data science.
Smooth embeddings of the Long Line and other non-paracompact manifolds into locally convex spacesmath.DG
We show that every finite dimensional Hausdorff (not necessarily paracompact, not necessarily second countable) -manifold can be embedded into a weakly complete vector space, i.e. a locally convex topological vector space of the form for an uncountable index set and determine the minimal cardin…
Study shows curvature constraints force submanifolds to have specific topology or geometry.
problem Curvature constraints on submanifolds in nonnegative curvature spaces.
method Investigates submanifolds with lower bounds on sectional curvature and mean curvature.
result Curvature constraints force submanifolds to have specific topology or geometry.
A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the o…
Diffeological vector spacesmath.DG
We study the relationship between many natural conditions that one can put on a diffeological vector space: being fine or projective, having enough smooth (or smooth linear) functionals to separate points, having a diffeology determined by the smooth linear functionals, having fine finite-dimensional subspaces, and hav…
Defines smooth actions of a group on manifolds and vector spaces.
problem Representing the general linear group and its actions.
method Restricted functor of points and category theory.
result Smooth actions on -graded vector spaces and manifolds.
Paper presents voxel graph operators for vector data models.
problem Efficient conversion and analysis of geometric models.
method Topological voxelization, graph construction, differential operator derivation.
result Discrete differential and integral operators from voxel complexes.
Topology guidance controls generative model outputs by specifying topological features.
problem Lack of control in generative models for visual analysis.
method Coupling a coordinate-based neural network with a diffusion model to guide sampling.
result Generated fields adhere to specified topological features while staying within the generative distribution.
In this article we consider the continuity of the eigenvalues of the connection Laplacian of -connections on vector bundles over Riemannian manifolds. To show it, we introduce the notion of the asymptotically -equivariant measured Gromov-Hausdorff topology on the space of metric measure spaces with isometric -…
The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a …
The paper develops algorithms and topological invariants for distinguishing dynamic systems.
problem Distinguishing the topological type of surfaces and functions in dynamic systems.
method Construction of algorithms and topological invariants using discrete topological structures.
result The development of discrete topological structures for topological equivalence of dynamic systems.
Generic singularities of line fields have been studied for lines of principal curvature of embedded surfaces. In this paper we propose an approach to classify generic singularities of general line fields on 2D manifolds. The idea is to identify line fields as bisectors of pairs of vector fields on the manifold, with re…
Classifies equivariant vector bundles over toric manifolds.
problem Classifying vector bundles over toric manifolds.
method Klyachko-type classification over invariant affine charts.
result Generalizes Klyachko's classification of toric vector bundles.
In this short note we prove that the degree of the Gauss map ν of a closed 3-dimensional hypersurface of the Euclidean space is a lower bound for the total bending functional B, introduced by G. Wiegmink. Consequently, the energy functional E introduced by C. M. Wood admits a topological lower bound.
We prove that smooth 1-dimensional topological field theories over a manifold are equivalent to vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism category, which encodes cutting laws rather than gluing laws. We make this idea precise through a smooth version of Rezk'…
Robust topological information commonly comes in the form of a set of persistence diagrams, finite measures that are in nature uneasy to affix to generic machine learning frameworks. We introduce a fast, learnt, unsupervised vectorization method for measures in Euclidean spaces and use it for reflecting underlying chan…
We classify orthogonal actions of finite groups on Euclidean vector spaces for which the corresponding quotient space is a topological, homological or Lipschitz manifold, possibly with boundary. In particular, our results answer the question of when the underlying space of an orbifold is a manifold.
Extends Loday-Quillen-Tsygan theorem to bornological Lie algebra homology.
problem Calculating Lie algebra homology of gauge algebras using cyclic homology.
method Extends proof to bornological Lie algebra homology of Fréchet and LF-algebras, prepares statements about homological algebra of topological vector spaces.
result Constructs a spectral sequence to calculate stable part of bornological Lie algebra homology of gauge algebras.
Approximates measures on curved spaces using Dirac measures.
problem Topology of invariant measures on curved manifolds.
method Introducing weakly regular vectors and approximating measures by Dirac measures.
result Ergodicity is a generic property in the space of invariant measures supported on weakly regular vectors.
We prove a Frobenius theorem for Banach distributions on manifolds that are modelled over locally convex spaces. Moreover, we recall how Frobenius theorems can be applied to infinite-dimensional Lie groups and obtain, that given a Lie subalgebra of the Lie algebra of a Lie group that is modelled over a locally convex s…
The paper explores Parseval frames on vector bundles, proving their existence for certain cases.
problem Existence of Parseval frames on vector bundles.
method Using -bundles and algebraic topology, the authors prove the existence of Parseval frames for orientable vector bundles and provide conditions for smaller size frames.
result The existence of Parseval frames for orientable vector bundles and conditions for smaller size frames.
A note on measure homologymath.GT
Measure homology was introduced by Thurston in order to compute the simplicial volume of hyperbolic manifolds. Berlanga endowed measure homology with a structure of graded locally convex (possibly non-Hausdorff) topological vector space. In this note we completely characterize Berlanga's topology on measure homology of…