The study explores finite group actions on elliptic 3-manifolds.
problem Finite, fiber-preserving group actions on elliptic 3-manifolds.
method General construction and restriction to elliptic 3-manifolds, proof for orientation-reversing and fiber-preserving diffeomorphisms, analysis of each type of elliptic 3-manifold.
result A presentation for the quotient space under finite group actions is constructed.
The study identifies exceptions to fiber-preserving symmetry in ODEs and systems.
problem Identifying exceptions to fiber-preserving symmetry in ODEs and systems.
method Lie's classification of Lie algebras of vector fields, absolute and relative scalar differential invariants, conditional and vector-valued relative invariants, prolongations of actions.
result Examples of scalar ODEs and systems with symmetry groups not fiber-preserving.
Study fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
problem Characterize fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
method Construct a class of involutions, extend product involutions, and use double covering.
result Any fiber-preserving, orientation-reversing involution factors as a product of an orientation-preserving and a specific class of involutions.
The paper studies finite group actions on Seifert manifolds, proving conditions for their existence.
problem Finite group actions on Seifert manifolds that preserve fiber and orientation.
method Constructs group actions and refines obstruction conditions.
result Provides general structure of finite groups that can act on Seifert manifolds.
In a paper published in 2002, the author gave a criterion to determine whether there is a fiber-preserving branched covering between two given orientable Seifert manifolds with orientable bases. Here we supply some details of the proof of two claims in that paper. We give an explicit construction of fiber-preserving br…
Paper solves equivalence problems for fifth-order differential operators using Cartan's method.
problem Equivalence problem for fifth-order differential operators under fiber-preserving transformations.
method Cartan's method of equivalence applied to solve two versions of the equivalence problem.
result Sufficient and necessary conditions for fiber-preserving transformations between fifth-order differential operators.
Paper extends group actions on Seifert manifolds with non-orientable base space.
problem Finite, fiber- and orientation-preserving group actions on orientable Seifert manifolds with non-orientable base space.
method Constructing a double cover and isomorphism between fiber-preserving diffeomorphisms and actions on the cover.
result Construction of all actions satisfying a condition on the obstruction term and structure of finite groups.
This article is dedicated to solve the equivalence problem for two third order differential operators on the line under general fiber--preserving transformation using the Cartan method of equivalence. We will do three versions of the equivalence problems: first via the direct equivalence problem, second equivalence pro…
In present paper, the equivalence problem for fourth order differential operators with one variable under general fiber-preserving transformation using the Cartan method of equivalence is applied. Two versions of equivalence problems are considered. First, the direct equivalence problem and second equivalence problem i…
Around 1960, R. Palais and J. Cerf proved a fundamental result relating spaces of diffeomorphisms and imbeddings of manifolds: If V is a submanifold of M, then the map from Diff(M) to Imb(V,M) that takes f to its restriction to V is locally trivial. We extend this and related results into the context of fibered manifol…
Study focuses on classifying special geometric structures.
problem Classify singular affine structures of integrable systems.
method Classification through simple semitoric systems equivalence.
result Counterexamples exist for multiple pinched fibers.
Maps between circle bundles are studied, proving fiber-preserving and finiteness results for mapping degrees.
problem Understanding the properties of maps between circle bundles and their degrees.
method Analyzing the structure of circle bundles over aspherical manifolds and using homotopy and homology properties.
result The mapping degree set of fiber-preserving maps from E1 to E2 is determined and finite under certain conditions. We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants…
We construct a family of split signature Einstein metrics in four dimensions, corresponding to particular classes of third order ODEs considered modulo fiber preserving transformations of variables.
This paper is devoted to apply the equivariant moving frame method to study the local equivalence problem of third order ordinarily differential equation under the pseudo-group of fiber preserving transformations.
Paper introduces 'zippers' for constructing universal circles.
problem Constructing universal circles for hyperbolic 3-manifolds.
method Introduces zippers to directly construct universal circles.
result New and direct way to construct universal circles.
Study local equivalence of Riemannian submersions using differential invariants.
problem Local equivalence problem for Riemannian submersions under fiber-preserving isometries.
method Analysis of differential invariants for orbit submersions induced by a Killing field.
result Explicit formulas for A and H in terms of base data (gˉ,φ,Ω) and equivalence criterion. The Nielsen Conjecture for Homeomorphisms asserts that any homeomorphism f of a closed manifold is isotopic to a map realizing the Nielsen number of f, which is a lower bound for the number of fixed points among all maps homotopic to f. The main theorem of this paper proves this conjecture for all orientation pre…
Study on braid monodromy of Lefschetz fibrations, proving infinite index subgroup.
problem Understanding the structure of braid monodromy in Lefschetz fibrations.
method Developed methods to show infinite index of subgroup, relating to character varieties.
result Proved infinite index subgroup for various cases of Lefschetz fibrations.
Let F′,F be any two closed orientable surfaces of genus g′>g≥1, and f:F→F be any pseudo-Anosov map. Then we can "extend" f to be a pseudo-Anosov map f′:F′→F′ so that there is a fiber preserving degree one map M(F′,f′)→M(F,f) between the hyperbolic surface bundles. Moreover the extension f′ can…
Classifies almost-toric systems in four dimensions.
problem Classifying compact almost-toric systems in four dimensions.
method Symplectic classification using base, Taylor series, and twisting indices.
result Classification into five invariants, analogous to semitoric systems.
In this paper we investigate for further symmetry properties of the nonlinear fin equations of the general form ut=(E(u)ux)x+h(x)u rather than recent works on these equations. At first, we study the projective (fiber-preserving) symmetry to show that equations of the above class can not be reduced to linear equa…
We show that on a closed smooth manifold M equipped with k fiber bundle structures whose vertical distributions span the tangent bundle, every smooth diffeomorphism f of M sufficiently close to the identity can be written as a product f=f1...fk, where fi preserves the ith-fiber. The factors …
Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometricians. Here we define a Riemannian or pseudo-Riemannian lift metric on TM, which is in some senses more…
Theory is developed for linear-quadratic at infinity generating families for Legendrian knots in R^3. It is shown that the unknot with maximal Thurston--Bennequin invariant of -1 has a unique linear-quadratic at infinity generating family, up to fiber-preserving diffeomorphism and stabilization. From this, invariant ge…
Let (M,g) be a smooth Riemannian manifold, K a compact Lie group and p:P→M a principal K-bundle over M endowed with a connection A. Fixing a bi invariant inner product on Lie algebra k of K, the connection A and metric g define a Riemannian metric gA on P. Let X~ be the …
A fibration of a Riemannian manifold is fiberwise homogeneous if there are isometries of the manifold onto itself, taking any given fiber to any other one, and preserving fibers. Examples are fibrations of Euclidean n-space by parallel n-planes, and the Hopf fibrations of the round n-sphere by great n-spheres. In this …
Classifies braid elements in elliptic fibrations up to conjugacy.
problem Classifying braid elements in elliptic fibrations up to conjugacy.
method Relating conjugacy classes to points on the SL2-character variety and norms on Z[ω] and Z[i].
result No elements of order n-1 or n-2 in Br(π).
Study on Ricci solitons on tangent and unit tangent bundles.
problem Characterizing Ricci solitons on tangent and unit tangent bundles.
method Analyzing pseudo-Riemannian g-natural metrics and their Ricci soliton properties. result Classification of conformal vector fields and existence of non-Einstein Ricci solitons.
New symmetries found for scalar and vector ODEs of arbitrary dimensions.
problem Identifying symmetries for scalar and vector ODEs of arbitrary dimensions.
method Explicit expressions and abelian Lie algebra for non-Cartan symmetries in arbitrary dimensions.
result Non-Cartan symmetries characterize linearizable systems of ODEs but not nonlinear ones.
Heinz Hopf's famous fibrations of the 2n+1-sphere by great circles, the 4n+3-sphere by great 3-spheres, and the 15-sphere by great 7-spheres have a number of interesting properties. Besides providing the first examples of homotopically nontrivial maps from one sphere to another sphere of lower dimension, they all share…
The elliptic 3-manifolds are the closed 3-manifolds that admit a Riemannian metric of constant positive curvature, that is, those that have finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic 3-manifold M, the inclusion from the isometry group of M to the diffeomorphism group of M…
Examples show cyclic actions on 4-manifolds can't extend to Hamiltonian circle actions.
problem Cyclic group actions on 4-manifolds that trivialize homology cannot be extended to Hamiltonian circle actions.
method Holomorphic methods applied to combinatorial tools for circle actions.
result Cyclic actions on 4-manifolds that trivialize homology do not extend to Hamiltonian circle actions.
Reduces proper actions to simpler core actions for analysis.
problem Understanding properties of proper actions on manifolds.
method Extending Skjelbred and Straume's construction to non-compact groups, focusing on core of actions.
result Properties of proper actions are determined by simpler core actions.
A new method learns disentangled macro actions from sequences for reinforcement learning.
problem Curse of dimensionality in reinforcement learning action space.
method Autonomously learns disentangled factor representation of actions to generate macro actions.
result Higher scores in complex environments compared to other reinforcement learning algorithms.
Introduces Conditional Action Trees to simplify RL action spaces.
problem Challenges in RL with large, complex action spaces.
method Structures action spaces and reduces complexity through Conditional Action Trees.
result Demonstrates effectiveness in reducing action space and improving decision making.
Study polar actions on Damek-Ricci spaces, proving existence and finding examples.
problem Characterize polar actions on Damek-Ricci spaces.
method Prove criteria for isometric actions to be polar, find examples, and classify actions.
result Non-trivial polar actions exist on all Damek-Ricci spaces.
New RL algorithm tackles complex discrete action spaces.
problem Challenges in applying on-policy RL in high-dimensional discrete action spaces.
method Action-value critic, correlated actions, gradient sparsification.
result Empirically outperforms related on-policy algorithms.
Finite group actions on smooth 3-manifolds can be smoothed.
problem Finite group actions on smooth 3-manifolds.
method Uniform limit of smooth actions.
result Every continuous action of a finite group on a smooth 3-manifold is a uniform limit of smooth actions.
AE-DQN learns to eliminate sub-optimal actions in complex RL environments.
problem Challenges in RL with many actions, especially redundancy.
method Combines DQN with AEN that predicts invalid actions.
result Significant speedup and robustness in games with many actions.
Reduction principles for proper actions on smooth manifolds.
problem Proper actions on smooth manifolds and their properties.
method Exhibit constructions and prove reduction principles for proper actions.
result Reduction principles hold for proper actions, polar actions, and copolarity.
Totally geodesic sections found in polar actions.
problem Understanding sections of polar actions on Riemannian manifolds.
method Elementary proof of a folklore result.
result Sections of polar actions are totally geodesic.
Simplifies large action space bandits by selecting representative actions.
problem Efficiently managing large action spaces with correlated outcomes.
method Random sampling and solving of bandit instances to identify representative actions.
result The algorithm selects a smaller set of representative actions that perform nearly as well as the full action space.
We study isometric actions on Riemannian symmetric spaces of noncompact type which are induced by reductive algebraic subgroups of the isometry group. We show that for such an action there exists a corresponding isometric action on a dual compact symmetric space, which reflects many properties of the original action. F…
The study proves conditions for symplectic torus actions on manifolds with non-contractible orbits.
problem Conditions for symplectic torus actions with non-contractible orbits.
method Analyzes symplectic torus actions on manifolds, proving conditions for Hamiltonian actions and orbit properties.
result Symplectic Tn−1 actions with non-contractible orbits are not Hamiltonian unless the orbits are contractible. Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.
problem Classifying totally geodesic submanifolds and polar actions on Stiefel manifolds.
method Classification through polar actions and cohomogeneity-one actions.
result Classification of orbits of polar actions on Stiefel manifolds.
Conditions for reducing quasi-actions to tree actions and group properties.
problem Conditions for reducing quasi-actions to tree actions.
method Reduction to cobounded isometric actions on trees.
result Groups with quasi-orbits quasi-isometric to trees are virtually free.
New stability theorem for non-hyperbolic group actions.
problem Structural stability of non-hyperbolic group actions.
method Introducing 'meandering hyperbolicity' for group actions on geodesic metric spaces.
result Meandering-hyperbolic actions are structurally stable.