Ideal triangulations of 3-manifolds are shown equivalent up to certain moves.
problem Equivalence of ideal triangulations in 3-manifolds.
method Using branched triangulations and transit equivalences, the paper shows that ideal triangulations are equivalent up to certain moves.
result Ideal triangulations of 3-manifolds are equivalent up to certain moves.
This paper establishes an equivalence between transitive double Lie algebroids and core diagrams.
problem Understanding and characterizing transitive double Lie algebroids.
method Using core diagrams and equivalence of transitive core diagrams with transitive double Lie groupoids.
result Transitive double Lie algebroids are completely determined by their core diagrams.
Proves spectra equivalence for Riemannian manifolds.
problem Equivalence of Almgren-Pitts and phase-transition half-volume spectra.
method Proof of spectra equivalence for Riemannian manifolds.
result Confirms conjecture about spectra equivalence.
The equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a well known result. A remarkable generalisation of this equivalence, due to Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admi…
This paper is a continuation of Part I where the general setup was developed. Here we discuss the general equivalence problem for geometric structures and provide criteria for the equivalence, local and global, of transitive structures. Cartan's Flag Systems illustrate the theory as a major example and, finally, some a…
Proves transitivity of a specific class of quadratic polynomials.
problem Transitivity of pure Hurwitz classes of post-critically finite quadratic polynomials.
method Uses mapping classes of the sphere with finitely many marked points.
result Establishes transitivity for pure Hurwitz classes of post-critically finite quadratic polynomials.
A key question in Reinforcement Learning is which representation an agent can learn to efficiently reuse knowledge between different tasks. Recently the Successor Representation was shown to have empirical benefits for transferring knowledge between tasks with shared transition dynamics. This paper presents Model Featu…
New relation on paths is not transitive.
problem Extending tree-like property to non-Lipschitz paths.
method Analyzing a fractal construction in the plane.
result The resulting relation is not an equivalence relation.
The paper studies triangulations of surfaces up to branched transit equivalences, proving equivalence conditions and foliations.
problem Understanding triangulations of surfaces up to branched transit equivalences.
method Analyzes triangulations of closed surfaces S with vertices V, considering branched triangulations up to b-transit equivalence generated by b-flips.
result Branched triangulations are equivalent under certain conditions, including parity of Euler-Poincare' characteristic c(S).
Due to a result by Mackenzie, extensions of transitive Lie groupoids are equivalent to certain Lie groupoids which admit an action of a Lie group. This paper is a treatment of the equivariant connection theory and holonomy of such groupoids, and shows that such connections give rise to the transition data necessary for…
We describe an algorithm for the enumeration of (candidates of) vertex-transitive combinatorial d-manifolds. With an implementation of our algorithm, we determine, up to combinatorial equivalence, all combinatorial manifolds with a vertex-transitive automorphism group on n≤13 vertices. With the exception of act…
Develops L∞ spaces over dg manifolds and establishes an equivalence with L∞ algebroids.
problem Defining and comparing L∞ spaces and algebroids over dg manifolds. method Establishes an equivalence between categories of L∞ algebroids and L∞ spaces, constructs a faithful functor. result Detects weak equivalences between L∞ algebroids and L∞ spaces. We classify multiply transitive homogeneous real (2,3,5) distributions up to local diffeomorphism equivalence.
We show that various notions of local homogeneity for CR-manifolds are equivalent. In particular, if germs at any two points of a CR-manifold are CR-equivalent, there exists a transitive local Lie group action by CR-automorphisms near every point.
The existence of nonconstant harmonic Dirichlet functions on a Cayley graph of a discrete group is equivalent to the nonvanishing of the first L2-cohomology of the given group. It was first proven by Cheeger and Gromov that such functions do not exists on the Cayley-graph of an amenable group. The result was extended u…
Paper explores state-action equivalence in RL, improving regret bounds.
problem Improving reinforcement learning performance by leveraging state-action equivalence.
method Introduces a notion of similarity between state-action pairs, defines equivalence structure, and presents algorithms for confidence sets.
result Confidence sets improve RL performance, especially in known equivalence structures.
New findings reveal discount regularization can be seen as a strong prior, leading to poor performance in unevenly sampled data.
problem Discount regularization leads to poor performance in unevenly sampled data.
method Equivalence theorem showing discount regularization as a strong prior, setting regularization parameters locally for individual state-action pairs.
result Discount regularization can be seen as a strong prior, leading to poor performance in unevenly sampled data.
Classifies multiply-transitive (2,3,5)-distributions using modern Cartan geometry.
problem Classifying multiply-transitive (2,3,5)-distributions. method Modern Cartan-geometric approach, incorporating G2 structure theory. result Complete classifications in both complex and real settings, with full curvature and infinitesimal holonomy.
RFMs transition from linear to nonlinear under specific input-label correlation.
problem Understanding the transition from linear to nonlinear behavior in RFMs.
method Analyzing RFMs under spiked covariance designs, characterizing the interaction between anisotropy and input-label correlation.
result The RFM generalization error is governed by the strength of input-label correlation, leading to a clear nonlinear advantage above a specific boundary.
This paper classifies expanding attractors and non-transitive Anosov flows on specific knot and manifold spaces.
problem Classifying expanding attractors and non-transitive Anosov flows on specific knot and manifold spaces.
method Using the derived Anosov (DA) expanding attractor and the Franks-Williams manifold, the paper proves the uniqueness of these structures.
result The DA expanding attractor and the Franks-Williams non-transitive Anosov flow are the unique structures supported by N0 and M0 respectively. We show that a self orbit equivalence of a transitive Anosov flow on a 3-manifold which is homotopic to identity has to either preserve every orbit or the Anosov flow is R-covered and the orbit equivalence has to be of a specific type. This result shows that one can remove a relatively unnatural assumption…
We classify the polar actions on the complex hyperbolic plane up to orbit equivalence. Apart from the trivial and transitive polar actions, there are five polar actions of cohomogeneity one and four polar actions of cohomogeneity two.
Elie Cartan's general equivalence problem is recast in the language of Lie algebroids. The resulting formalism, being coordinate and model-free, allows for a full geometric interpretation of Cartan's method of equivalence via reduction and prolongation. We show how to construct certain normal forms (Cartan algebroids) …
Study of Morita equivalences in Lie groupoids and their symmetries.
problem Exploring Morita equivalences in Lie groupoids.
method Investigation of principal bibundles and Morita equivalences in Pfaffian groupoids.
result Introduction and analysis of Pfaffian Morita equivalence.
This research reinterprets Lie and Cartan's work on geometric structures using Lie groupoids.
problem Revisiting Lie and Cartan's geometric structures from a modern perspective.
method Encoding geometric structures into principal G-bundles with a transversally parallelisable foliation. result Developed a notion of flatness for Lie groupoids encompassing various geometric structures.
Classifies pseudo-Anosov flows on 3-manifolds up to orbit equivalence.
problem Classifying pseudo-Anosov flows on 3-manifolds up to orbit equivalence.
method Generalized Anosov-like actions on bifoliated planes, ideal boundary analysis.
result Pseudo-Anosov flows on 3-manifolds are determined up to orbit equivalence by their ideal boundary actions.
Horizontal surgery on pseudo-Anosov flows yields almost equivalent flows.
problem Understanding and manipulating pseudo-Anosov flows.
method Performing horizontal surgery on pseudo-Anosov flows by cutting along specific annuli and regluing with a Dehn twist.
result Horizontal Goodman surgery on transitive pseudo-Anosov flows yields an almost equivalent flow.
Paper proposes a new Markov model for efficient PLC system design.
problem Efficient estimation of Markov model parameters for bursty error channels.
method Introduced a Block Diagonal Markov model and a modified Baum-Welch algorithm.
result Efficient estimation of state transition matrix Λ for PLC system design. Researchers discover all affinely homogeneous models for surfaces in 4D space.
problem Identifying all affinely homogeneous models for surfaces in 4D space.
method Improved power series method of equivalence, capturing invariants at the origin, creating branches, and infinitesimalizing calculations.
result Find several inequivalent terminal branches yielding each to some nonempty moduli space of homogeneous models.
One of the longstanding open problems in spectral graph clustering (SGC) is the so-called model order selection problem: automated selection of the correct number of clusters. This is equivalent to the problem of finding the number of connected components or communities in an undirected graph. We propose automated mode…
The Jacobian conjecture is simplified using polynomial mappings.
problem Simplifying the Jacobian conjecture over the real field.
method Using polynomial mappings to restrict transitions on manifolds.
result An equivalent statement of the Jacobian conjecture.
We prove a general uniformization theorem for N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces, showing that in general an N=2 superconformal (resp. N=1 superanalytic) DeWitt super-Riemann surface is N=2 superconformally (resp., N=1 superanalytically) equivalent to a manifold with transition funct…
New insights into pseudo-Anosov flows with special periodic orbits.
problem Understanding pseudo-Anosov flows with periodic orbits in 3-manifolds.
method Analyzing the topological features corresponding to trees of scalloped regions and classifying flows with the same free homotopy data.
result Explicit examples of flows with the same free homotopy data but not orbit equivalent.
We study surfaces with decorations and prove uniformization in non-Euclidean geometries.
problem Discrete conformal equivalence in non-Euclidean geometries.
method Variational principle and continuous deformation.
result One master theory of discrete conformal equivalence across different geometries.
We classify all transitive actions of Lie algebras of vector fields on C^3 and R^3 up to a local equivalence and discuss why this classification can not be extended in general to the solvable case. The main technical tool is the structure of one-dimensional invariant foliations on homogeneous spaces.
Paper reviews algebraic research in machine learning theory.
problem Understanding phase transitions in machine learning models.
method Algebraic approaches in statistical mechanics.
result Algebraic methods are essential for analyzing machine learning models with singularities.
EPIC quantifies reward differences without policy optimization.
problem Distinguishing reward function quality from policy optimization issues.
method EPIC distance to compare reward functions directly.
result EPIC bounds policy training success and regret.
Unified understanding of integrability obstructions for Lie algebroids.
problem Equivalence of integrability obstructions for transitive Lie algebroids.
method Unified approach using cohomological and homotopical methods.
result Unified understanding of integrability obstructions.
New method shows pseudo-Anosov flows on graph manifolds can be simplified.
problem Understanding pseudo-Anosov flows on graph manifolds.
method Constructing a partial Birkhoff section with genus one components that misses finitely many closed orbits.
result Every pseudo-Anosov flow on a graph manifold is almost equivalent to a totally periodic flow or a suspension Anosov flow.
Model place cells as spatial embeddings for efficient path planning and cognitive map construction.
problem Encoding spatial navigation in the hippocampus.
method Model place cells using spectral decomposition of multi-step random walk transition kernels, inducing sparsity and adjacency.
result Place cells encode spatial information through non-negativity and inner-product structure, forming a cognitive map.
New loss function equivalence reveals PER's uniform sampling can be improved.
problem Improving Prioritized Experience Replay (PER) for better learning efficiency.
method Transforming non-uniformly sampled data loss functions into uniformly sampled ones.
result Some environments can replace PER with a new loss function without performance loss.
Given a linearly ordered set I, every surjective map p: A --> I endows the set A with a structure of set of preferences by "replacing" the elements of I with their inverse images via p considered as "balloons" (sets endowed with an equivalence relation), lifting the linear order on A, and "agglutinating" this structure…
MuZero visualizes its internal representations to stabilize planning.
problem Stability issues in MuZero's planning process.
method Visualized MuZero's latent representations and proposed regularization techniques.
result Action trajectories diverge between observation embeddings and internal state transitions, leading to instability.
Modeling financial markets as gas molecules, the paper predicts phase transitions similar to water and steam.
problem Understanding the dynamics of financial markets through phase transitions.
method Developed a lattice gas model equivalent to the Ising model on a social network, analyzing critical exponents and auto-correlations.
result Financial market dynamics exhibit phase transition-like behavior, with critical exponents analogous to water and steam.
Study of flows on complex manifolds with holomorphic properties.
problem Global rigidity of transversely holomorphic Anosov flows on smooth compact manifolds.
method Analyzing the integrability of unstable and stable distributions, proving uniqueness in low dimensions.
result For topologically transitive flows, they are either orbit equivalent to a hyperbolic automorphism or geodesic flow.
New abelian cohomology theory applied to rigidity of flows.
problem Rigidity of Anosov flows and negatively curved surfaces.
method Development of abelian Livshits theory for transitive Anosov flows.
result Abelian Livshits theorem for homologically full Anosov flows.
This paper shows how post-Lie algebra structures can be induced by simply transitive NIL-affine actions.
problem Understanding which solvable Lie groups can act simply transitively on nilpotent Lie groups.
method Introducing post-Lie algebra structures and showing their correspondence with simply transitive actions.
result Simply transitive NIL-affine actions induce complete post-Lie algebra structures in the 2-step nilpotent case.
We consider branes N in a Schwarzschild-AdS(n+2) bulk, where the stress energy tensor is dominated by the energy density of a scalar fields map $\f:N\ra \mc S$ with potential V, where $\mc S$ is a semi-Riemannian moduli space. By transforming the field equation appropriately, we get an equivalent field …