This paper determines the flexible exponent for non-geometric 3-manifolds.
problem Bounding the mapping degree in terms of the Lipschitz constant for non-geometric 3-manifolds.
method Analyzing the infimum of α such that the inequality holds for any Lipschitz map.
result The flexible exponent for non-geometric 3-manifolds is determined.
Flexible approach for normal approximations in geometric and topological statistics.
problem Normal approximation for complex statistics not expressible as sums of score functions.
method Flexible add-one cost operator combined with strong stabilization theory.
result Established normal approximation results for geometric and topological statistics.
Introduces a new geometric method for optimal experimental design.
problem Restrictive invariance properties of traditional OED approaches based on probability densities.
method Mutual transport dependence (MTD) using optimal transport theory.
result Demonstrates high-quality designs and flexibility compared to standard methods.
In this paper we study geometric, algebraic, and computational aspects of flexibility and infinitesimal flexibility of Kokotsakis meshes. A Kokotsakis mesh is a mesh that consists of a face in the middle and a certain band of faces attached to the middle face by its perimeter. In particular any 3x3-mesh made of quadran…
We show that surface groups are flexibly stable in permutations. This is the first non-trivial example of a non-amenable flexibly stable group. Our method is purely geometric and relies on an analysis of branched covers of hyperbolic surfaces. Along the way we establish a quantitative variant of the LERF property for s…
We describe the cohomology ring of the moduli space of a flexible polygon in geometrically meaningful terms. We propose two presentations, both are computation friendly: there are simple rules for cup product.
New bounds on mapping degrees for geometric 3-manifolds.
problem Bounding the mapping degree in terms of Lipschitz constant for geometric 3-manifolds.
method Constructing Legendrian maps to prove bounds on flexible exponent.
result Complete result for flexible exponent of geometric 3-manifolds.
Characterizes geometric actions on graphs with flexible stabilizers.
problem Understanding geometric actions on flexible stabilizers.
method Defining generalized fine actions and proving relative quasi-convexity criteria.
result Characterizes Bowditch boundary points in relatively geometric actions.
This paper concerns the questions of flexibility and rigidity of solutions to the Monge-Ampère equation which arises as a natural geometrical constraint in prestrained nonlinear elasticity. In particular, we focus on anomalous i.e. "flexible" weak solutions that can be constructed through methods of convex integration …
Given a flexible n-gon with generic side lengths, the moduli space of its configurations in R2 as well as in R3 is a smooth manifold. It is equipped with n \textit{tautological} line bundles whose definition is motivated by M. Kontsevich's tautological bundles over M0,n. We st…
We explain a connection between the algebraic and geometric properties of groups of contact transformations, open book decompositions, and flexible Legendrian embeddings. The main result is that, if a closed contact manifold (V,ξ) has a supporting open book whose pages are flexible Weinstein manifolds, then the conn…
Geometric families of low-rank covariances improve flexibility and tractability in high dimensions.
problem Interpolating and identifying covariance matrices in high dimensions with limited data.
method Differential geometric construction of low-rank covariance families, interpolation on manifolds, and distance minimization for identification.
result Differential geometric covariance families offer significant flexibility and computational tractability.
New method for flexible tubes and structures, enabling rigid-foldability.
problem Creating flexible tubes with rigid-foldability.
method Discrete, semi-discrete, and smooth construction of surfaces (T-hedra and profile-affine surfaces).
result Unified treatment of continuous flexible structures composed of tubes.
Quantum neural networks need both data-dependent and trainable unitaries for effective geometric deformation.
problem Quantum neural networks lack the geometric flexibility of classical networks due to limitations in state reachability.
method Viewing quantum states as embedded manifolds, we analyze infinitesimal unitary actions and introduce the CLA maps and aCLS criterion.
result Geometric flexibility in quantum neural networks requires a joint dependence on data and trainable weights.
Groups of importance in group theory have flexible stability properties.
problem Stability and flexibility of groups in geometric and combinatorial group theory.
method Establishing Kirchberg's Local Lifting Property and Lubotzky--Shalom's Property FD for specific groups.
result Groups like 3-manifold groups, limit groups, and certain one-relator groups are very flexibly stable. We solve the Plateau problem for marginally outer trapped surfaces in general Cauchy data sets. We employ the Perron method and tools from geometric measure theory to force and control a blow-up of Jang's equation. Substantial new geometric insights regarding the lower order properties of marginally outer trapped surfa…
We prove estimates interpolating the Schwarz Lemmata of Royden-Yau and the ones recently established by the author. These more flexible estimates provide additional information on (algebraic) geometric aspects of compact Kähler manifolds with nonnegative holomorphic sectional curvature, nonnegative $\Ric_\ell$ or posit…
The Weierstrass representation for minimal surfaces in R3 provides a flexible method for constructing minimal surfaces of arbitrary genus. The topological limitations of minimal surfaces interfere with this providing a more general geometric modeling tool. Minimal surfaces lie in the larger class of harmoni…
Consider a smooth closed surface M of fixed genus ⩾2 with a hyperbolic metric σ of total area A. In this article, we study the behavior of geometric and dynamical characteristics (e.g., diameter, Laplace spectrum, Gaussian curvature and entropies) of nonpositively curved smooth metrics with total area …
Randomized control methods improve asset pricing and performance analysis.
problem Challenges in drawing inferences from traditional random portfolios in performance evaluation.
method Geometric random walks and Markov chain Monte Carlo methods to construct flexible control groups.
result Captured premia associated with size, value, quality, and momentum in a constrained setting.
Geometric approach combines asset returns and investor views for better portfolio optimization.
problem Optimizing portfolios with investor-specific views.
method Generalized Wasserstein barycenter (GWB) to integrate statistical asset returns and investor views.
result The geometric approach offers more flexibility and rewards for correct investor views.
We propose a flexible convex relaxation for the phase retrieval problem that operates in the natural domain of the signal. Therefore, we avoid the prohibitive computational cost associated with "lifting" and semidefinite programming (SDP) in methods such as PhaseLift and compete with recently developed non-convex techn…
Flexible spatial models improve predictive performance over nonstationary alternatives.
problem Improving predictive performance in nonstationary spatial modeling.
method Introduces a modular parametric covariance function that extends nonstationary spatial models.
result The proposed covariance function outperforms nonparametric methods in predictive performance.
Geodesic curves improve flexibility in covariance estimation.
problem Inflexible covariance families limit spatiotemporal modeling.
method Use geodesic curves to build more flexible covariance families.
result Natural projection minimizes geodesic distance to sample covariance.
Proves small cancellation free products have geometric actions on CAT(0) cube complexes.
problem Proving small cancellation free products have geometric actions on CAT(0) cube complexes.
method Using a blown-up complex of groups and a boundary separation criterion, proving wall stabilizers form a rich family of subgroups.
result Proves $C'( rac16)$--small cancellation free products of residually finite groups are residually finite.
Deep learning models complex multivariate extremes using geometric shapes.
problem Modeling complex extremal dependencies in high-dimensional data.
method Geometric representation and deep learning for flexible semi-parametric models.
result First approach to modeling limit sets using deep learning for high-dimensional data.
This paper summarizes closed-form relations for SE(3) maps and their derivatives.
problem Closed-form expressions for SE(3) maps and their derivatives are scattered in the literature.
method Summarizes and provides proofs for relevant closed-form relations of the exponential and Cayley map on SE(3).
result Provides an implicit generalized-alpha scheme for rigid/flexible multibody systems using the Cayley map.
Extends diffusion models to non-Euclidean spaces with geometric priors.
problem Difficulties in natural sciences with symmetries and non-Euclidean data.
method Constructs a noising process and neural network equivariant to symmetry group, approximates score function.
result Model can generate complex scalar and vector fields on synthetic and real-world data.
We give a general criterion for the Dirichlet problem at infinity (DPI) on a Cartan-Hadamard surface to be solvable, which we primarily use to give the best possible upper radial radial curvature bound for solvability of the DPI, but which is also flexible enough to accommodate flats. In particular, any (upper) radial …
Neural ODEs extended to manifolds for flexible sampling.
problem Sampling from complex multimodal distributions on non-trivial topologies.
method Extending Neural ODEs to smooth manifolds using vector fields.
result A general methodology for building normalizing flows on manifolds.
Braid groups are an important and flexible tool used in several areas of science, such as Knot Theory (Alexander's theorem), Mathematical Physics (Yang-Baxter's equation) and Algebraic Geometry (monodromy invariants). In this note we will focus on their algebraic-geometric aspects, explaining how the representation the…
Proves upper bounds for heat kernels evolving on manifolds.
problem Bounding heat kernels on evolving manifolds.
method Logarithmic Sobolev inequalities and ultracontractivity estimates.
result Gaussian upper bounds for heat kernels are derived.
Motivated by the flexibility of biological neural networks whose connectivity structure changes significantly during their lifetime, we introduce the Unstructured Recursive Network (URN) and demonstrate that it can exhibit similar flexibility during training via gradient descent. We show empirically that many of the di…
Extends geometric approach to model non-stationary extremal dependence.
problem Capturing evolving extremal dependence in multivariate data.
method Geometric framework for non-stationary multivariate extreme value modelling.
result Framework can capture various dependence forms and is robust to different model formulations.
Paper introduces a new geometric homology theory and applies it to Gromov-Witten theory.
problem Developing a new homology theory for orbifolds with corners.
method Using stratification and triangulation theories of Lie groupoids and their orbit spaces, extending to Lie groupoids with corners.
result Proposes and proves the geometric homology theory (GHT), a flexible generalization of singular homology.
New framework reveals limits of flexible, periodic thin surfaces.
problem Understanding the mechanical behavior of thin, periodic surfaces.
method Developed a duality between surface rotations and in-plane stresses.
result Exactly three out of six possible strain states are isometries.
Study of polygon spaces, characterizing critical points of area function.
problem Characterizing critical points of area function in polygon spaces.
method Geometric characterization of critical points and calculation of Morse indices.
result Generalization of isoperimetric theorems for polygons in the plane.
In this paper we define and study flexible links and flexible isotopy in projective space. Flexible links are meant to capture the topological properties of real algebraic links. We classify all flexible links up to flexible isotopy using Ekholms interpretation of Viros encomplexed writhe.
The authors characterize flexibility in power and energy markets considering time, spatiality, resource, and risk.
problem Evaluating and maximizing flexibility in power systems and markets.
method Characterization of flexibility dimensions (time, spatiality, resource, risk) and their interrelations with flexibility assets, products, and services.
result Flexibility should be evaluated based on multiple dimensions for efficient power systems and markets.
New inequalities for convex hypersurfaces in various spaces.
problem Deriving inequalities for hypersurfaces under convex weight.
method Sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth, closed hypersurfaces in Euclidean, spherical, and hyperbolic spaces.
result Incorporates convex, non-decreasing positive functions as weights, yielding a broad family of geometric inequalities.
Study presents a twistor correspondence for specific geometric structures.
problem Twistor theory for almost-Grassmannian manifolds.
method Utilizes moduli of curves-with-boundary for global correspondence.
result Foundational results in complex setting, global correspondence for real Grassmannian.
Generative models improve angular variable simulation in high dimensions.
problem Lack of flexibility and scalability in simulating multivariate angular variables.
method Introducing generative adversarial networks, normalizing flows, and flow matching.
result Deep learning methods outperform classical parametric models in complex data structures.
New neural method calculates EMD for particle physics data.
problem Metric for particle collider events based on Wasserstein metric.
method Neural network architecture estimating EMD using Kantorovich-Rubinstein duality.
result Differentiable way to calculate EMD for geometric fitting.
Many open problems and important theorems in low-dimensional topology have been formulated as statements about certain 2--complexes called gropes. This paper describes a precise correspondence between embedded gropes in 4--manifolds and the failure of the Whitney move in terms of iterated `towers' of Whitney disks. The…
New inequality for odd-degree flexible curves using surface doubling.
problem Bounding the number of non-empty ovals of odd-degree flexible curves.
method Defining an Arnold surface for odd-degree flexible curves and using it to derive a Viro--Zvonilov-type inequality.
result Upper bound on the number of non-empty ovals of odd-degree flexible curves.
Learning workable representations of dynamical systems is becoming an increasingly important problem in a number of application areas. By leveraging recent work connecting deep neural networks to systems of differential equations, we propose \emph{variational integrator networks}, a class of neural network architecture…
Automates detection of fast-ramped flexibility events for DSOs.
problem Monitoring and supervising flexibility activations in power systems.
method Unsupervised detection and open-set classification.
result Automatically identifies critical flexibility activations for early intervention.
Flexible surfaces found in complex projective and product spaces.
problem Finding flexible surfaces in complex projective and product spaces.
method Constructing flexible surfaces within prescribed homology classes.
result Flexible surfaces exist in both CP2 and S2imesS2.