The paper adapts results for Reeb flows and Hamiltonian flows, showing all orbits are closed have identical periods.
problem Adapting results for Reeb flows and Hamiltonian flows with closed orbits.
method Adapting results from Geodesic circle foliations to Reeb and Hamiltonian flows.
result All orbits on connected contact manifolds with closed orbits have identical periods.
The study finds infinitely many periodic orbits that can be used to modify Anosov flows.
problem Can surgeries on periodic orbits of Anosov flows produce equivalent flows?
method Analyzing suspension Anosov flows, the study identifies pairs of periodic orbits that can be used to modify the flow.
result For some suspension Anosov flows, there exist infinitely many pairs of periodic orbits that can be used to modify the flow.
Study Vassiliev invariants and periodic orbits of Axiom A flows.
problem Calculating Vassiliev invariants and writhe for periodic orbits of Axiom A flows.
method Asymptotic analysis of Vassiliev invariants and writhe.
result Obtained asymptotics for Vassiliev invariants and writhe of periodic orbits.
The paper classifies periodic solitons in curve flows on the light-cone.
problem Investigating periodic solitons in curve flows on the light-cone.
method Deriving Harnack inequality for heat flow, classifying space-periodic solitons for a third-order curvature flow.
result Closed soliton solutions form a family of transcendental curves with specific rotation indices.
Study shows no periodic geodesics in jet space.
problem Existence of periodic geodesics in jet space.
method Characterization and classification of subRiemannian geodesics.
result No periodic geodesics found in the space of k-jets.
The space of non-singular flows on any given solenoid is shown to contain a generic subset consisting of flows that are not almost periodic. Whether this result carries over to Hamiltonian flows remains an open question.
The study finds dense orbits and absolute period leaves for complex flows.
problem Existence of dense orbits for real Rel flows on holomorphic 1-forms.
method Established a density criterion for mSL(2,R)-orbit closures, verified using explicit constructions. result Found dense leaves and examples of absolute period foliation.
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.
New proof confirms periodic orbit conjecture for Eulerisable flows.
problem Periodic orbit conjecture for non-vanishing vector fields on closed manifolds.
method Characterization of Eulerisable flows and use of strongly adapted one-forms.
result Periodic orbit conjecture holds for Eulerisable flows.
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.
Paper shows how to transform certain flows into R-covered ones.
problem Transforming Anosov flows into R-covered ones.
method Goodman-Fried surgery along periodic points.
result Any topologically transitive Anosov flow can be transformed into an R-covered one.
The author shows that equicontinuous geodesic flows on surfaces are periodic. A similar result for flows on 3-manifolds is also proven. The idea of the proof is to show that the return map is recurrent and therefore periodic.
Anosov flows' orbit complements are hyperbolic manifolds.
problem Characterizing hyperbolicity of Anosov flow orbit complements.
method Analyzing foliations and periodic orbits in 3D manifolds.
result Complement of any filling periodic orbit in Anosov flows is hyperbolic.
Paper defines the payback period for nonconventional cash flows using axioms.
problem Defining the payback period for nonconventional cash flows is challenging.
method Used axiomatic approach to define the payback period.
result The last break-even point of the project balance is the only definition consistent with axioms.
Geometrically solves Schrödinger flow on sphere.
problem Solving periodic Cauchy problem for Schrödinger flow on sphere.
method Explicit geometric algorithm for construction of solutions.
result Explicit geometric algorithm for solving Schrödinger flow on sphere.
For Hamiltonian flows we establish the existence of periodic orbits on a sequence of level sets approaching a Bott-nondegenerate symplectic extremum of the Hamiltonian. As a consequence, we show that a charge on a compact manifold with a nondegenerate (i.e. symplectic) magnetic field has periodic orbits on a sequence o…
The paper generalizes index theory for periodic manifolds and proves equivalence of signatures for tori.
problem Generalizing index theory for periodic manifolds and proving signature equivalence for tori.
method Periodic index theory and spectral flow for elliptic complexes, surgery formula for singular instanton homology.
result Equivalence of signatures for essentially embedded tori and surgery formula for singular instanton homology.
The main result of this article is that if a 3-manifold M supports an Anosov flow, then the number of conjugacy classes in the fundamental group of M grows exponentially fast with the length of the shortest orbit representative, hereby answering a question raised by Plante and Thurston in 1972. In fact we show th…
The Whitham flow for hyperelliptic curves has singularities that can be smoothly extended.
problem Singularities in the Whitham flow for hyperelliptic spectral curves.
method Analysis of deformations preserving periods of a meromorphic differential.
result Stable and unstable manifolds are non-empty, and the flow can be extended through the singularity.
Nguyen's solutions converge to a grim reaper and plane.
problem Classifying semigraphical translators for mean curvature flow.
method Constructing a one-parameter family of translating solutions.
result Nearly complete classification of semigraphical translators.
The study finds sufficient conditions for Reeb flows to have genus zero global surfaces of section.
problem Finding conditions for Reeb flows to have genus zero global surfaces of section.
method Analyzes linking assumptions on periodic orbits and ambient contact geometry.
result Reveals sufficient conditions for Reeb flows to have genus zero global surfaces of section.
Study flat flow solutions to Mullins-Sekerka and area-preserving curvature flows on planar flat torus.
problem Behavior of flat flow solutions on planar flat torus.
method Sharp quantitative Alexandrov inequality derivation for periodic smooth sets.
result Flat flows converge to specific configurations exponentially fast.
New findings on magnetic geodesic flows and periodic motions.
problem Characterizing superintegrable systems in magnetic geodesic flows.
method Analyzing rotationally symmetric magnetic geodesic flows.
result All sufficiently slow motions in a central magnetic field are periodic under specific curvature and homogeneity conditions.
We construct new examples of self-translating surfaces for the mean curvature flow from a periodic configuration with finitely many grim reaper cylinders in each period. Because this work is an extension of the author's article on the desingularization of a finite family of grim reaper cylinders, we simply discuss the …
New insights into Reeb dynamics from Finsler geometry.
problem Understanding periodic orbits in Reeb flows.
method Contact geometry, Hamiltonian circle actions, surgery construction.
result Compact manifolds with arbitrarily large numbers of periodic Reeb orbits.
Study on periodic orbits in contact geometry and Finsler geodesics.
problem Existence and non-existence of periodic Reeb orbits on contact manifolds.
method Interpretation of Finsler geodesics as Reeb flows and use of open books.
result Existence of periodic Reeb orbits on contact manifolds with suitable open books.
Hamiltonian dynamical systems tend to have infinitely many periodic orbits. For example, for a broad class of symplectic manifolds almost all levels of a proper smooth Hamiltonian carry periodic orbits. The Hamiltonian Seifert conjecture is the existence problem for regular compact energy levels without periodic orbits…
A new method for valuing insurance liabilities using cost-of-capital approach.
problem Valuation of insurance liabilities with market-consistency and cost considerations.
method Two-stage valuation: replicate liability cash flow first, then manage residual cash flow with capital constraints.
result Explicit formulas and properties of the cost-of-capital margin under specific assumptions.
Integrable flows on null curves in anti-de Sitter 3-space studied.
problem Analyzing integrable flows on null curves in anti-de Sitter 3-space.
method Formulated integrable flows related to the KdV hierarchy on null curves exploiting the geometry of anti-de Sitter 3-space.
result Explicitly found closed stationary solutions in terms of periodic solutions of a Lamé equation.
The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems. We show that almost all low levels of a function on a geometrically bounded symplectically aspherical manifold carry contractible periodic orbits of the Hamiltonian flow, provided th…
Study of straight-line flows on a unique infinite surface.
problem Understanding straight-line flows on a specific infinite surface.
method Geometric description and characterization of periodic and drift orbits; use of rigid symmetries and Veech group.
result Complete characterization of periodic directions and proof of density of periodic and ergodic directions.
Explains how Reeb dynamics connects to topology.
problem Understanding the relationship between Reeb dynamics and topology.
method Contact cuts, lifting group actions, and other construction methods.
result Methods for constructing Reeb flows with finite periodic orbits.
Consider a mean curvature flow of hypersurfaces in Euclidean space, that is initially graphical inside a cylinder. There exists a period of time during which the flow is graphical inside the cylinder of half the radius. Here we prove a lower bound on this period depending on the Lipschitz-constant of the initial graphi…
We classify the periodic Hamiltonian flows on compact four dimensional symplectic manifolds up to isomorphism of Hamiltonian S^1 spaces. Additionally, we show that all these spaces are Kaehler, that every such space is obtained from a simple model by a sequence of symplectic blowups, and that if the fixed points are is…
We show that if K: P \to R is an autonomous Hamiltonian on a symplectic manifold (P,Ω) which attains 0 as a Morse-Bott nondegenerate minimum along a symplectic submanifold M, and if c_1(TP)|_M vanishes in real cohomology, then the Hamiltonian flow of K has contractible periodic orbits with bounded period on all suffici…
SRMF with high ESG ratings outperform during economic crises.
problem Performance of socially responsible mutual funds during economic downturns.
method Comparative analysis of SRMF with different ESG ratings.
result High ESG rated SRMF outperform low ESG rated SRMF during economic crises.
We give examples of rank one compact surfaces on which there exist recurrent geodesics that cannot be shadowed by periodic geodesics. We build rank one compact surfaces such that ergodic measures on the unit tangent bundle of the surface are not dense in the set of probability measures invariant by the geodesic flow. F…
Geometric proof for averaging theorem on Riemannian manifolds.
problem Averaging theorem for perturbed dynamical systems on Riemannian manifolds.
method Geometric proof using a free coordinate approach.
result Generalization of averaging procedure to any open domain with compact closure.
The present paper is a review of counterexamples to the ``Hamiltonian Seifert conjecture'' or, more generally, of examples of Hamiltonian systems having no periodic orbits on a compact energy level. We begin with the discussion of the ``classical'' and volume--preserving Seifert conjectures. Then a construction of coun…
This paper uses Heegaard Floer theory to study pseudo-Anosov flows and their periodic points.
problem Understanding the differential of Heegaard Floer chain complexes associated with pseudo-Anosov flows.
method Introduces a refined grading to analyze the homology of subcomplexes representing irreducible multi-orbits.
result The homology of these subcomplexes is 1-dimensional, providing insights into periodic points of pseudo-Anosov flows.
Improved LSTM for industrial flow prediction with higher accuracy.
problem Predicting continuous time series variables in industrial settings.
method LSTM algorithm with multivariate tuning, incorporating periodic measurement and time window concepts.
result Significantly improved prediction accuracy (54.05% higher than traditional LSTM).
In this article we analyze totally periodic pseudo-Anosov flows in graph three manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show …
We study the determination of the second-order normal form for perturbed Hamiltonians Hε=H0+εH1+2ε2H2, relative to the periodic flow of the unperturbed Hamiltonian H0. The formalism presented here is global, and can be easily implemented in any CAS. We illustrate it by means of two examples: the H…
In this article, we study the knots realized by periodic orbits of R-covered Anosov flows in compact 3-manifolds. We show that if two orbits are freely homotopic then in fact they are isotopic. We show that lifts of periodic orbits to the universal cover are unknotted. When the manifold is atoroidal, we deduce some fin…
Geometric flow on curves in S^3 generates YO equations solutions.
problem Modeling short wave-long wave interaction.
method Simple geometric flow on curves in S3. result Constructs transverse curves for YO equations periodic solutions.
We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration, determining its asymptotic behavior.
Study periodic orbits in specific parts of surface moduli spaces.
problem Understanding periodic orbits in the thin part of strata.
method Analyzing Teichmueller flow on specific subsets of moduli spaces.
result Asymptotic growth rate of periodic orbits in thin parts is at least h(Q)-1.
Study on mean curvature flow in a cone, proving existence and homogenization.
problem Mean curvature flow in a cone with periodic boundary conditions.
method Construction of self-similar solutions, a priori estimates, homogenization limit analysis.
result Global existence of radially symmetric solutions and characterization of the homogenization limit.