Study FRS deformations of space curves with focus on flattenings, vertices, and twistings.
problem Geometric analysis of space curves with specific deformations.
method Investigation of FRS-deformations of space curves with emphasis on geometric properties. result Detailed study of flattenings, vertices, and twistings points on space curves.
Let G be a compact connected simple Lie group and let $M=G^{\bb{C}}/P=G/K$ be a generalized flag manifold. In this article we focus on an important invariant of G/K, the so called $\fr{t}$-root system $R_{\fr{t}}$, and we introduce the notion of symmetric $\fr{t}$-triples, that is triples of $\fr{t}$-roots $ξ, ζ, η…
A new method for estimating large-scale linear models with improved precision.
problem Estimating large-scale linear statistical models efficiently.
method Sequential Least-Squares Estimators with Fast Randomized Sketching (SLSE-FRS), integrating Sketch-and-Solve and Iterative-Sketching methods.
result SLSE-FRS produces high-precision estimators, outperforming state-of-the-art methods.
Study shows splitting schemes can approximate WFR flows faster than the exact flow.
problem Improving sampling efficiency in Wasserstein-Fisher-Rao gradient flows.
method Investigates operator splitting techniques to numerically approximate WFR flows.
result A judicious choice of step size and operator ordering can lead to faster convergence of split schemes to the target distribution.
FR-Train improves fair and robust AI training by detecting and reducing poisoned data.
problem Training AI models that are fair and robust in the presence of data bias and poisoning.
method Mutual information-based adversarial training with an additional discriminator.
result FR-Train maintains fairness and accuracy even in the presence of poisoned data.
Proves FR-NGD optimally approximates evolutionary dynamics and continuous Bayesian inference.
problem Optimizing continuous time replicator equations and continuous Bayesian inference.
method Fisher-Rao natural gradient descent (FR-NGD) and its correspondence with evolutionary dynamics.
result FR-NGD optimally approximates continuous time replicator equations and continuous Bayesian inference.
Fuzzy Rough Set selects key features for phishing detection.
problem Efficient feature selection for phishing detection.
method Fuzzy Rough Set (FRS) theory for feature selection.
result Maximum F-measure of 95% using Random Forest.
FR-LUX optimizes portfolio management by learning cost-aware policies robust to market conditions.
problem Transaction costs and regime shifts cause failure in live trading portfolios.
method Integrates three ingredients: microstructure-consistent execution model, trade-space trust region, and explicit regime conditioning.
result Achieves top average Sharpe ratio, maintains flat cost-performance slope, and superior risk-return efficiency.
This study provides an explicit expansion of KL divergence's gradient flow in Fisher-Rao geometry.
problem Sampling techniques struggle to traverse between modes in non-convex potential functions.
method Explicit expansion of KL divergence's gradient flow in Fisher-Rao geometry.
result The convergence rate to π is independent of the potential function.
We recall an extension of Kirby's Calculus on non-simply connected 3-manifolds given in [FR], and the surgery calculus of bridged links from [Ke], which involves only local moves. We give a short combinatorial proof that the two calculi are equivalent, and thus describe the same classes of 3-manifolds. This makes the p…
Universal construction for Lie algebroids from anchored bundles with connections.
problem Creating a universal framework for Lie algebroids from anchored bundles with connections.
method Adapting Kapranov's construction to anchored bundles with arbitrary connections, showing compatibility with Lie algebroid structure.
result Universal connection ildeabla on FR(E) compatible with Lie algebroid structure, turning (FR(E),ildeabla) into a Cartan-Lie algebroid. Paper proposes FR algorithm to solve minimax optimization locally.
problem Gradient descent fails to find local minimax in minimax optimization.
method Follow-the-Ridge (FR) algorithm, addressing rotational behavior of gradient dynamics.
result FR algorithm provably converges to local minimax.
Study geodesics in homogeneous spaces and find conditions for their existence.
problem Characterize geodesics in homogeneous spaces.
method Analyze geodesics of the form γ(t)=π(exp(tX)exp(tY)) in G/K. result Existence of geodesics for specific conditions in homogeneous spaces.
We study geodesics in generalized Wallach spaces which are expressed as orbits of products of three exponential terms. These are homogeneous spaces M=G/K whose isotropy representation decomposes into a direct sum of three submodules m=m1⊕m2⊕m3, satisfying the relations $[\fr…
This article follow the article {http://hal.archives-ouvertes.fr/hal-00361030/fr/} in which the author characterize the fact of being of finite volume for a convex projective surface. We show here that the moduli space βf(Σg,p) of the convex projective structure on the surface Σg,p of genius g with p pun…
A new category generates 1D tangle invariants.
problem Developing a new category for 1D tangles.
method Proving a new (∞,1)-category has universal mapping property. result The new category generates link invariants.
New method improves fairness of facial recognition systems.
problem Facial recognition systems exhibit bias across different demographic groups.
method Optimizes centroid-based scores to reduce bias in pre-trained models.
result Demonstrates significant improvement in fairness with minimal loss in accuracy.
Study on cosymplectic groupoids with structural results.
problem Understanding cosymplectic groupoids and their properties.
method Structural analysis of cosymplectic groupoids, comparison with other groupoids.
result Several structural results for cosymplectic groupoids.
Study reveals centralization in Bitcoin transactions involving retail users.
problem Centralization and bias in Bitcoin transaction data.
method Heuristic classification of Bitcoin users, weekly activity pattern analysis.
result Most real transactions involve Frequent Receivers, centralizing the ecosystem.
Paper defines PB-groupoids and their relation to VB-groupoids.
problem Establishing PB-groupoids as a Lie 2-groupoid counterpart.
method Defining PB-groupoids as Lie 2-groupoid actions on diagrams of Lie groupoids and principal bundles.
result PB-groupoids generalize VB-groupoids and provide a new correspondence between them and principal bundles.
The paper studies groupoid structures from different viewpoints.
problem Understanding groupoid morphisms and their structures.
method Constructing two groupoids from morphisms of groupoids, one from a categorical viewpoint and the other from a geometric viewpoint. Showing equivalence of the two kinds of groupoids of morphisms.
result Equivalence of two kinds of groupoids of morphisms for each pair of groupoids.
The paper generalizes bundle gerbes over groupoids and their correspondence with PB groupoids.
problem Generalizing bundle gerbes over groupoids and their properties.
method Developed a functorial correspondence between PB groupoids and bundle gerbes over groupoids.
result Built a correspondence between PB groupoids and bundle gerbes over groupoids, including partial quotients.
We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger's fundamental groupoid from the fundamental double groupoid of a Lie groupoid. In the case of a symplectic double groupoid, we study the induced closed 2-form on the associated l…
A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all groupoid structure maps are continuous. The notion of monodromy groupoid of a topological groupoid generalises those of fundamental groupoi…
Lie Calculus connects differential and Lie theory using groupoids.
problem Understanding the relationship between differential and Lie theories.
method Using groupoids to link differential and Lie theories.
result Higher order theory involves higher algebra (n-fold groupoids).
We outline the construction of the holonomy groupoid of a locally Lie groupoid and the monodromy groupoid of a Lie groupoid. These specialise to the well known holonomy and monodromy groupoids of a foliation, when the groupoid is just an equivalence relation.
We introduce the notions of a differentiable groupoid and a differentiable stratified groupoid, generalizations of Lie groupoids in which the spaces of objects and arrows have the structures of differentiable spaces, respectively differentiable stratified spaces, compatible with the groupoid structure. After studying b…
We construct examples of finitely generated groups L that have non-trivial actions on R-trees but which cannot act, without fixing a vertex, on any simplicial tree. Moreover, any finitely presented group mapping onto L does have a fixed point-free action on some simplicial tree.
This work examines curvature effects on empirical mean in Riemannian and affine manifolds for small samples.
problem Understanding curvature effects on empirical mean in small samples on Riemannian and affine manifolds.
method Established Taylor expansions for the first and second moments of the empirical mean, showing curvature impacts.
result Explicit formulas for bias and modulation of empirical mean's covariance matrix due to curvature.
We complete the construction of the double Lie algebroid of a double Lie groupoid begun in the first paper of this title. We show that the Lie algebroid structure of an LA--groupoid may be prolonged to the Lie algebroid of its Lie groupoid structure; in the case of a double groupoid this prolonged structure for either …
We discuss two sorts of generalization of Lie groupoids. One is Lie n-groupoids defined as simplicial manifolds with trivial πk≥n+1. The other is the stacky Lie groupoid $\cG\rra M$ with $\cG$ a differentiable stack. We build 1-1 correspondence between Lie 2-groupoids and stacky Lie groupoids up to a certain…
Improved action recognition in live videos with hybrid FR-DL method.
problem High computational costs and lack of temporal information in conventional action recognition.
method Automated selection of representative frames, feature extraction, background subtraction, HOG, deep neural network, LSTM, Softmax-KNN classifier.
result Significant improvement in accuracy and speed compared to state-of-the-art methods.
In this work we deal with coverings and actions of Lie group- groupoids being a sort of the structured Lie groupoids. Firstly, we define an action of a Lie group-groupoid on some Lie group and the smooth coverings of Lie group-groupoids. Later, we show the equivalence of the category of smooth actions of Lie group-grou…
We introduce the notion of Glanon groupoids, which are Lie groupoids equipped with multiplicative generalized complex structures. It combines symplectic groupoids, holomorphic Lie groupoids and holomorphic Poisson groupoids into a unified framework. Their infinitesimal, Glanon Lie algebroids are studied. We prove that …
Paper solves Hilbert's fifth problem for specific groupoids.
problem When are transitive groupoids continuously isomorphic to Lie groupoids?
method Investigates proper and transitive groupoids with compact source fibers.
result Solves Hilbert's fifth problem for specified groupoids.
Introduces multiplicative differential forms on Lie groupoids with VB-groupoids values.
problem Describing multiplicative differential forms on Lie groupoids with VB-groupoids values.
method Introduces multiplicative differential forms on Lie groupoids with values in VB-groupoids, presents a Lie theory for differential forms on Lie groupoids with values in 2-term representations up to homotopy, defines a differential complex whose 1-cocycles are multiplicative forms with values in VB-groupoids.
result Complete description of multiplicative differential forms on Lie groupoids with values in VB-groupoids.
Solves open problem on Lie groupoids equivalence.
problem Whether Lie groupoids Morita equivalent are diffeologically Morita equivalent.
method Localisation of 2-categories, anafunctors, Lie groupoids, diffeological groupoids.
result Two Lie groupoids diffeologically Morita equivalent are Morita equivalent in the Lie sense.
The paper extends Hilbert's fifth problem to transitive groupoids.
problem When is a transitive topological groupoid continuously isomorphic to a Lie groupoid?
method Investigation of transitive topological groupoids, focusing on proper and transitive groupoids with compact source fibers.
result Presentation of results generalizing Hilbert's fifth problem to transitive groupoids, including solutions for proper and transitive groupoids with compact source fibers.
New dataset for urban point cloud segmentation and classification.
problem Lack of high-quality urban point cloud datasets for machine learning.
method Acquired by Mobile Laser Scanning (MLS), post-processed and labeled.
result Dataset can be used for learning classification and segmentation algorithms.
Pycobra is a Python toolbox for ensemble learning and visualization.
problem Ensemble learning and visualization for machine learning tasks.
method Implementation of ensemble learning algorithms and a flexible interface for blending any machine learning algorithm.
result Visualisation tools like Voronoi tessellations for ensemble learning.
In this note a functorial approach to the integration problem of an LA-groupoid to a double Lie groupoid is discussed. To do that, we study the notions of fibred products in the categories of Lie groupoids and Lie algebroids, giving necessary and sufficient conditions for the existence of such. In particular, it turns …
Analyzes how suborbifolds relate to groupoid embeddings.
problem Understanding the relationship between suborbifolds and groupoid embeddings.
method Examines the correspondence between suborbifolds and groupoid embeddings via atlases and effective orbifold groupoids.
result Identifies classes of suborbifolds that naturally lead to groupoid embeddings.
We approach Mackenzie's LA-groupoids from a supergeometric point of view by introducing Q-groupoids, which are groupoid objects in the category of Q-manifolds. There is a faithful functor from the category of LA-groupoids to the category of Q-groupoids. We associate to every Q-groupoid a double complex that provides a …
In this paper we study the Lie groupoids which appear in foliation theory. A foliation groupoid is a Lie groupoid which integrates a foliation, or, equivalently, whose anchor map is injective. The first theorem shows that, for a Lie groupoid G, the following are equivalent: - G is a foliation groupoid, - G has discrete…
The article proves Lie's third theorem for Lie algebroids via singular Lie groupoids.
problem Lie's third theorem does not hold for Lie groupoids and Lie algebroids.
method Introducing a subcategory of diffeological spaces called quasi-etale, constructing a functor mapping singular Lie groupoids to Lie algebroids.
result Lie's third theorem is valid for Lie algebroids within the context of singular Lie groupoids.
Locally conformal symplectic (l.c.s.) groupoids are introduced as a generalization of symplectic groupoids. We obtain some examples and we prove that l.c.s. groupoids are examples of Jacobi groupoids in the sense of \cite{IM}. Finally, we describe the Lie algebroid of a l.c.s. groupoid.
The study examines fibrations of Lie groupoids and their classification.
problem Understanding and classifying fibrations of Lie groupoids.
method Introduced locally topological product Lie groupoid fibrations and used groupoid cohomology to obstruct and classify these fibrations.
result The existence of locally topological product Lie groupoid fibrations is obstructed by a specific groupoid cohomology.
Paper defines weak representations and their equivalence to VB-groupoids.
problem Understanding representations of Lie groupoids and VB-groupoids.
method Introduces weak representations and shows equivalences between categories.
result Equivalence between 2-term representations up to homotopy and weak representations of G.