Wise's Quasiconvex Hierarchy Theorem classifying hyperbolic virtually compact special groups in terms of quasiconvex hierarchies played an essential role in Agol's proof of the Virtual Haken Conjecture. Answering a question of Wise, we construct a new virtual quasiconvex hierarchy for relatively hyperbolic virtually co…
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48 results for “Hierarchy Structure Theorem”
Derive bihamiltonian structure for rational reduction of 2D-Toda hierarchy
problem Derive bihamiltonian structure for rational reduction of 2D-Toda hierarchy
method Direct computations
result Derive local bihamiltonian structure
The paper proves a hierarchy of Liouville theorems for polyharmonic functions on manifolds with nonnegative Ricci curvature.
problem Establishing Liouville theorems for polyharmonic functions on manifolds with nonnegative Ricci curvature.
method A new estimate for the Laplacian of a polyharmonic function, obtained by induction through a cutoff construction combined with a hole-filling argument.
result All polyharmonic functions of sublinear growth on manifolds of nonnegative Ricci curvature are constant.
Scroll structures on solutions of 4D integrable equations are involutive and governed by a dispersionless hierarchy.
problem Characterizing the geometry of solutions to 4D integrable equations.
method Defining rational normal scrolls and showing their involutivity.
result Involutive scroll structures are governed by a dispersionless integrable hierarchy.
Constructs tri-Hamiltonian structure and Frobenius manifold for asymmetric gAL hierarchy
problem Tri-Hamiltonian structure and Frobenius manifold for asymmetric gAL hierarchy
method Local tri-Hamiltonian structure construction and Frobenius manifold construction
result Dispersionless limits of flows belong to Principal Hierarchy
Starting from a so-called flat exact semisimple bihamiltonian structures of hydrodynamic type, we arrive at a Frobenius manifold structure and a tau structure for the associated principal hierarchy. We then classify the deformations of the principal hierarchy which possess tau structures.
This paper develops a cohomological hierarchy for bistable visual paradoxes.
problem Understanding the hierarchy of visual paradoxes built from bistable elements.
method Develops a cohomological hierarchy using coefficients and a discrete Stokes theorem.
result Reveals a hierarchy of paradox classes from through , refined at each degree by the relative/absolute distinction.
New integrable deformations for topological hierarchies from Frobenius manifolds.
problem Integrable deformations of topological hierarchies from Frobenius manifolds.
method Construction of integrable deformations with polynomial tau-structures.
result Conjecture of universal object for Riemann--Hopf hierarchy.
Local tri-Hamiltonian structure for Ablowitz-Ladik hierarchy established.
problem Establishing a tri-Hamiltonian structure for the Ablowitz-Ladik hierarchy.
method Constructing a local tri-Hamiltonian structure and computing central invariants.
result Central invariants of one bi-Hamiltonian structure are 1/24, and dispersionless limit matches Frobenius manifold.
New diffiety theory leads to a well-posed KP hierarchy.
problem Formalizing diffieties for better hierarchy analysis.
method Definition of diffiety based on Frolicher structures.
result Formation of a well-posed Kadomtsev-Petviashvili hierarchy.
Constructs integrable hierarchies for generalized Frobenius manifolds with non-flat unity.
problem Integrable hierarchies for generalized Frobenius manifolds with non-flat unity.
method Constructs a bihamiltonian integrable hierarchy of hydrodynamic type.
result Integrable hierarchy possesses Virasoro symmetries and a tau structure.
The study improves geometric estimates for surfaces of constant mean curvature in 3-manifolds.
problem Estimating geometric properties of surfaces with constant mean curvature in 3-manifolds.
method Application of the Hierarchy Structure Theorem to understand global properties of surfaces, including area and diameter.
result Improved geometric estimates for the area and diameter of surfaces with constant mean curvature, including genus bounds.
Paper investigates methods to improve classification by inducing a hierarchy from flat labels.
problem Improving classification performance on datasets lacking a natural hierarchy.
method The approach involves clustering conditional distributions and using a hierarchical classifier with the induced hierarchy.
result The methods can discover latent hierarchies and improve accuracy in various applications.
Study bihamiltonian structures and Frobenius manifolds for specific Toda hierarchies.
problem Local bihamiltonian structures and Frobenius manifolds for asymmetric rational reductions of 2D-Toda hierarchy.
method Construct three-dimensional generalized Frobenius manifold, relate to other hierarchies via transformations.
result Explicit relation between RR2T and bi-graded Toda and constrained KP hierarchies.
We compute the central invariants of the bihamiltonian structures of the constrained KP hierarchies, and show that these integrable hierarchies are topological deformations of their hydrodynamic limits.
We give an alternate proof of Wise's Malnormal Special Quotient Theorem (MSQT), avoiding cubical small cancellation theory. We also show how to deduce Wise's Quasiconvex Hierarchy Theorem from the MSQT and theorems of Hsu--Wise and Haglund--Wise.
We establish a rigorous link between infinite-dimensional regular Frölicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a version of the Kadomtsev-Petviashvili hierarchy on a regular Frölicher Lie group of series of non-formal odd-class pseudodi…
Super tau-covers extend bihamiltonian hierarchies' symmetries.
problem Extending symmetries of bihamiltonian hierarchies.
method Constructing super tau-covers for bihamiltonian integrable hierarchies.
result Symmetries of bihamiltonian hierarchies extended to super tau-covers.
Paper addresses limitations of traditional hierarchical clustering methods.
problem Traditional hierarchical clustering methods face limitations in binary trees and ultrametrics.
method Introduces the notion of a valid hierarchy and a two-step algorithm to construct a binary tree and prune it to enforce validity.
result Proposes a method to recover the finest valid hierarchy, which is not constrained to binary structures.
New method for deep learning hierarchies like sequences and graphs.
problem Lack of formalism for dealing with hierarchical structures in deep learning.
method Express equivariant maps for hierarchies using combinations of building block symmetries.
result Demonstrates effectiveness on semantic segmentation of point-cloud data.
We present some general results on properties of the bihamiltonian cohomologies associated to bihamiltonian structures of hydrodynamic type, and compute the third cohomology for the bihamiltonian structure of the dispersionless KdV hierarchy. The result of the computation enables us to prove the existence of bihamilton…
We construct integrable hierarchies of flows for curves in centroaffine through a natural pre-symplectic structure on the space of closed unparametrized starlike curves. We show that the induced evolution equations for the differential invariants are closely connected with the Boussinesq hierarchy, and …
We obtain variational formulas for holomorphic objects on Riemann surfaces with respect to arbitrary local coordinates on the moduli space of complex structures. These formulas are written in terms of a canonical object on the moduli space which corresponds to the pairing between the space of quadratic differentials an…
Study identifies pitfalls in assessing hierarchies for multi-class classification.
problem Lack of understanding in selecting hierarchies for multi-class classification.
method Analyzed and compared popular approaches to extracting hierarchies.
result Hierarchy quality becomes irrelevant when using powerful classifiers.
The hierarchy structure associated with a (2+1)-dimensional Nonlinear Schroedinger equation is discussed as an extension of the theory of the KP hierarchy. Several methods to construct special solutions are given. The relation between the hierarchy and a representation of toroidal Lie algebras are established by using …
The modular vector field of a Poisson-Nijenhuis Lie algebroid is defined and we prove that, in case of non-degeneracy, this vector field defines a hierarchy of bi-Hamiltonian -vector fields. This hierarchy covers an integrable hierarchy on the base manifold, which may not have a Poisson-Nijenhuis structure.
New tools in nonlinear random matrices improve understanding of the Sum of Squares hierarchy.
problem Improving the Sum of Squares (SoS) hierarchy's performance on average-case problems.
method Developed new tools in nonlinear random matrices and applied them to analyze the SoS hierarchy.
result Subexponential-time SoS lower bounds for various problems, offering evidence for the low-degree likelihood ratio hypothesis.
Paper proves vanishing terms in a second Poisson bracket for a specific system.
problem Proving polynomiality of coefficients in the dispersion parameter expansion of the second Poisson bracket.
method Bi-Hamiltonian recursion and Liu-Pandharipande relations.
result Proves vanishing terms in the second Poisson bracket expansion.
Proves efficient learning of hierarchical structure in meta-reinforcement learning.
problem Lack of provable guarantees for learning hierarchical structures in reinforcement learning.
method Analyzed HRL in meta-RL setting with tabular transition dynamics, providing diversity conditions and regret bounds.
result Sample-efficient recovery of natural hierarchical structure with provable guarantees.
A symmetry reduction of the Painlevé IV hierarchy to the Flaschka-Newell Painlevé II hierarchymath-ph
Symmetry reduction of Painlevé IV to Flaschka-Newell Painlevé II
problem Isomonodromic deformation problem associated with rank-two meromorphic connections
method Symmetry
result Induced isomonodromic dynamics coincides with Flaschka-Newell Painlevé II hierarchy
First geometric proof of the flyping theorem.
problem Proving Tait's flyping conjecture.
method Geometric proof using Greene's characterization, Menasco's crossing ball structures, and isotopy/re-plumbing moves.
result First entirely geometric proof of Menasco-Thistlethwaite's flyping theorem.
Building systems that autonomously create temporal abstractions from data is a key challenge in scaling learning and planning in reinforcement learning. One popular approach for addressing this challenge is the options framework (Sutton et al., 1999). However, only recently in (Bacon et al., 2017) was a policy gradient…
This work improves metric learning models by incorporating class hierarchies.
problem Class hierarchies are often ignored in classification-based metric learning models.
method Trained softmax classifier and metric learning models with predefined class hierarchies.
result ProxyDR model performs better in hierarchical inference and hierarchy-informed performance.
POSCMs extend SCMs for causal modeling with latent contexts.
problem Causal modeling with latent contexts and endogenous mechanisms.
method Kolmogorov-Arnold-Sprecher edge-functional decomposition for explicit parametrization.
result Identifiability of structure and mechanisms under latent context.
We propose an extension of the structure equation for constant mean curvature (CMC) surfaces in a three dimensional Riemannian space form to the associated CMC hierarchy of evolution equations by the higher-order commuting symmetries. Via the canonical formal Killing field, considered as an infinitely prolonged and loo…
New hierarchies and equations derived from Poisson structures.
problem Development of new hierarchies and equations from Poisson structures.
method Introduction and classification of multicomponent Poisson structures, derivation of hierarchies and equations.
result New Harry Dym and Hunter-Saxton equations derived for arbitrary number of components.
Ambrose-Singer theorem on diffeological bundles and complete integrability of the KP equationmath.DG
In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Frölicher Lie groups, state an Ambrose-Singer theorem that enlarges the one stated in \cite{Ma2}, and conclude with a differential geometric treatment of KP hierarchy. The example…
Investigates proving geometric theorems over complex and real numbers using tilings.
problem Proving incidence theorems over C and R using the master theorem.
method Formalizes tiling proofs and introduces a hierarchy of theorems based on topological spaces.
result Identifies which theorems can or cannot be proved over C and R.
Different from the traditional classification tasks which assume mutual exclusion of labels, hierarchical multi-label classification (HMLC) aims to assign multiple labels to every instance with the labels organized under hierarchical relations. Besides the labels, since linguistic ontologies are intrinsic hierarchies, …
Continuing the previous work, we propose a further extension of the structure equation for a truncated CMC hierarchy by the non-commuting, truncated Virasoro algebra of non-local symmetries. Via a canonical dressing transformation, we first define a wave function for the CMC hierarchy. This leads to a pair of additiona…
Large-scale classification of data where classes are structurally organized in a hierarchy is an important area of research. Top-down approaches that exploit the hierarchy during the learning and prediction phase are efficient for large scale hierarchical classification. However, accuracy of top-down approaches is poor…
New method generates critical points for complex functionals.
problem Proving the Willmore conjecture in complex geometric settings.
method Minmax hierarchies and fibrations for critical points.
result New proof of the Willmore conjecture.
We provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, this criteria characterizes relative hyperbolicity. We apply our c…
Similarity algebra extends algebraic structures with quantitative bounds.
problem Exact algebraic structures with strict axioms.
method Framework for approximate algebraic and Lie structures with -estimates.
result Similarity structures converge to classical algebraic objects as .
Researchers develop Orlov-Schulman symmetries for self-dual conformal structures.
problem Developing symmetries for self-dual conformal structures.
method Explicit proof of compatibility with Lax-Sato flows, dressing scheme based on Riemann-Hilbert problem.
result Construction and proof of compatibility of Orlov-Schulman symmetries.
Study on hypermaps and KP hierarchy, proving tau function and enumerative meaning.
problem Understanding the partition function of meromorphic functions on the Riemann sphere.
method Analysis of Hurwitz Dubrovin--Frobenius manifold structure and rational reductions of the KP hierarchy.
result The all genera partition function is a tau function of a rational reduction of the Kadomtsev--Petviashvili hierarchy.
An important problem in multi-label classification is to capture label patterns or underlying structures that have an impact on such patterns. This paper addresses one such problem, namely how to exploit hierarchical structures over labels. We present a novel method to learn vector representations of a label space give…
Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler spaces are applied in fractional calculus with Caputo derivatives and for elaborating models of fractional gravity and fractional Lagrange mechanics. The geometric data for such models are encoded into (fractional) bi-Hamiltonian structures and as…