The study connects lamination and orbit closures in hyperbolic manifolds.
problem Understanding the geometric and dynamical properties of horocycle orbit closures in Z-covers of compact hyperbolic manifolds. method Exposes connections between distance minimizing laminations and horospherical orbit closures in Z-covers of compact hyperbolic manifolds. Provides novel constructions and explicit descriptions. result Even slight perturbations to hyperbolic metrics can drastically change horocycle orbit closures.
We study a regular closure operator in the category of quandles. We show that the regular closure operator and the pullback closure operator corresponding to the reflector from the category of quandles to its full subcategory of trivial quandles coincide, we give a simple description of this closure operator, and analy…
The paper studies knitted surfaces and surface-links, showing their isotopy and closure properties.
problem Understanding the isotopy and closure properties of knitted surfaces and surface-links.
method Analyzing the structure and closure of knitted surfaces and surface-links in R4. result Any surface-link is ambient isotopic to the closure of a 2-dimensional knit.
New surfaces with special geodesic and horocycle behaviors discovered.
problem Understanding geodesic and horocycle dynamics on hyperbolic surfaces.
method Constructing geometrically infinite hyperbolic surfaces with tailored recurrence properties.
result First examples of non-trivial minimal horocyclic orbit closures and infinite locally-finite conservative horocyclic invariant measures.
It is known that there are 21 ribbon knots with 10 crossings or fewer. We show that for every ribbon knot, there exists a tangle that satisfies two properties associated with the knot. First, under a specific closure, the closed tangle is equivalent to its corresponding knot. Second, under a different closure, the clos…
Model distinguishes homophily and triadic closure in network analysis.
problem Conflated properties of homophily and transitivity in network analysis.
method Generative model with triadic closure edges and inference procedure.
result Can identify mechanisms responsible for network edges and community structure.
We prove results about orbit closures and equidistribution for the SL(2,R) action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows. The proofs of the main theorems rely on the measure classification theorem of [EMi2] and a certain isolation property of closed SL(2,R)…
A wealth-process set is abstractly defined to consist of nonnegative càdlàg processes containing a strictly positive semimartingale and satisfying an intuitive re-balancing property. Under the condition of absence of arbitrage of the first kind, it is established that all wealth processes are semimartingales and that t…
Constructs Lepage equivalents for arbitrary-order Lagrangians.
problem Creating Lepage equivalents for complex Lagrangians.
method Uses variational bicomplex and symmetric linear connections to construct Lepage equivalents satisfying the closure property.
result Shows how to extend global Lepage equivalents to ones satisfying the closure property.
The study explores knots and manifolds, proving properties and non-left-orderable groups.
problem Properties of knots and manifolds after surgeries and fillings.
method Fixed point method and Dehn surgery.
result Fundamental groups of certain manifolds are not left orderable.
We consider (compact or noncompact) Lorentzian manifolds whose holonomy group has compact closure. Among other results, we obtain that this property is equivalent to admitting a parallel timelike vector field. We also derive some properties of the space of all such metrics on a given manifold.
The paper proves properties of manifolds under certain conditions.
problem Understanding properties of manifolds under specific conditions.
method Analyzing homotopy cofibrations and rational elliptic closures.
result Almost all manifolds satisfy Moore's conjecture at sufficiently large primes.
Algorithm converts plat to standard closure of braids in 3D and related spaces.
problem Converting plat to standard closure of braids in different spaces.
method Algorithmic approach for plat to standard closure conversion in \(\mathbb{R}^3\), handlebodies, and thickened surfaces.
result Algorithm is quadratic for plat to standard closure and linear for standard to plat closure.
In this paper we aim at the description of foliations having tangent sheaf TF with c1(TF)=c2(TF)=0 on non-uniruled projective manifolds. We prove that the universal covering of the ambient manifold splits as a product, and that the Zariski closure of a general leaf of F is an…
Translation surfaces can be defined in an elementary way via polygons, and arise naturally in in the study of various basic dynamical systems. They can also be defined as Abelian differentials on Riemann surfaces, and have moduli spaces called strata that are related to the moduli space of Riemann surfaces. There is a …
Study on knot classification using 3-braid closures and ribbon surfaces.
problem Classifying smoothly slice knots from 3-braid closures.
method Construct ribbon surfaces and use twisted Alexander polynomial.
result Classification of knots up to 20 crossings.
New proof classifies orbit closures in Hodge bundle.
problem Classifying mGL+(2,R)-orbit closures in Hodge bundle. method Using deformations of flat pairs of pants.
result Short proof of absolute period foliation classification.
We prove that for a fixed braid index there are only finitely many possible shapes of the annular Rasmussen dt invariant of braid closures. Applying the same perspective to the knot Floer invariant ΥK(t), we show that for a fixed concordance genus of K there are only finitely many possibilities for ΥK(t). Fo…
Study orbits in right triangles, deducing periodic billiard paths and classifying orbit closures.
problem Understanding periodic billiard paths in right triangles and orbit closures in strata of Abelian and quadratic differentials.
method Classifying orbit closures of rank at least two in hyperelliptic components of strata of Abelian and quadratic differentials.
result Computed orbit closures and deduced asymptotic number of periodic billiard trajectories in right triangles.
Proves closure for specific spacetimes with certain conditions.
problem Proving closure for globally hyperbolic spacetimes.
method Using a Bonnet-Myers type result.
result Proves closure for spacetimes with specific conditions.
Method finds domain of Laplace-Beltrami operator on 2D almost-Riemannian manifolds.
problem Determining the domain of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds with tangency points.
method Using tools from Lie groupoids, natural domains of perturbations are found.
result Method allows treatment of geometries with tangency points.
In this paper, we establish a sufficient condition for a geodesic in a Riemannian manifold to be homogeneous, i.e. an orbit of an 1-parameter isometry group. As an application of this result, we provide a new proof of the fact that every weakly symmetric space is geodesic orbit manifold, i.e. all its geodesics are ho…
Classifies Zariski closures of positive representations in Lie groups.
problem Classifying Zariski closures of positive representations in Lie groups.
method Classifies the Lie algebra of the Zariski closure of a discrete subgroup with specific properties.
result Obtains a new proof of Guichard's classification of Zariski closures of Hitchin representations.
Study proposes curvature flow model for Drosophila dorsal closure.
problem Modeling and understanding Drosophila dorsal closure during embryonic development.
method Curvature-based mathematical model, analysis of maximum-principle and integral-estimates, numerical approximation scheme.
result Established global existence and convergence for the model.
Improved bounds on combining hypothesis classes for binary functions.
problem Understanding how to combine hypothesis classes for binary functions.
method Established upper bounds on Littlestone and threshold dimensions for combined classes.
result Upper bounds are nearly tight and give exponential improvements.
Adaptive rerouting reshapes impacts of maritime chokepoint disruptions
problem How disruptions to shipping traffic at chokepoints affect global economy
method Empirically calibrated full-scale agent-based model of global commercial shipping fleet
result Rerouting changes arrival losses under chokepoint closures
Classifies orbit closures in translation surface strata.
problem Classifying orbit closures in translation surface strata.
method Classification of extGL(2,R) orbit closures. result Applications to joinings of certain Masur-Veech measures.
Uniform Closure Method and Bayes classifier perform similarly in classifying open knots.
problem Classifying knots in open macromolecular chains.
method Used the Bayes MAP classifier and compared it to the Uniform Closure Method.
result Both methods have comparable accuracy and positive predictive value.
We discuss two different in general natural approaches to the ideal closure and ideal boundary of Busemann nonpositively curved metric space. It is shown that the identity map of the space admits surjective continuation from its coarse ideal closure to the weak one. We consider some situations when these closures coinc…
A new method predicts non-Markovian closure terms for complex systems.
problem Predicting the effect of unresolved variables on resolved dynamics in high-dimensional systems.
method Mamba-Assisted Closure (MAC) framework: sequence model trained to predict closure from resolved trajectory, coupled with reduced-order equations.
result Substantially outperforms existing methods in predictive accuracy and long-time stability.
The study bounds the number of closed geodesics in a specific orbit closure of surfaces.
problem Counting closed geodesics in a specific orbit closure of surfaces.
method Analyzes triangulations and Teichmüller geodesics to bound the number of closed geodesics.
result Obtains exponential bounds on the number of closed geodesics of length at most R.
Extends graph degree theorem to simplicial closure of Auter space.
problem Connectivity of graphs in Auter space.
method Defines degree for simplicial closure, extends Hatcher-Vogtmann theorem.
result Simplicial closure of Auter space is (d-1)-connected for degree d.
We study a certain type of braid closure which resembles the plat closure but has certain advantages; for example, it maps pure braids to knots. The main results of this note are a Markov-type theorem and a description of how Vassiliev invariants behave under this braid closure.
We give a Dehn-Nielsen type theorem for the homology cobordism group of homology cylinders by considering its action on the acyclic closure, which was defined by Levine, of a free group. Then we construct an additive invariant of those homology cylinders which act on the acyclic closure trivially. We also describe some…
The paper defines plat closures for spherical braids and shows links in RP3 can be realized this way.
problem Defining and analyzing plat closures for spherical braids in RP3. method Defining plat closures, associating residual permutations, and presenting moves on spherical braids.
result The number of components of the plat closure link of a spherical braid is equal to the number of disjoint cycles in its residual permutation.
We will say that a group G possesses the Magnus property if for any two elements u,v in G with the same normal closure, u is conjugate to v or v^{-1}. We prove that some one-relator groups, including the fundamental groups of closed nonorientable surfaces of genus g>3 possess this property. The analogous result for ori…
This article is a survey on Lorenz knots. We describe the original construction, prove several classical properties, in particular the fact that the closure of a positive braid is a fibered knot, and describe Ghys'correspondance between modular knots and Lorenz knots. We also prove two new properties, namely that follo…
The paper classifies orbit closures of symplectic Lie algebras.
problem Classifying orbit closures of symplectic Lie algebras under the action of Sp(4,R). method Analyzing the natural action of Sp(4,R) on the set of 4-dimensional Lie algebras with symplectic structures. result A complete classification of orbit closures of 4-dimensional symplectic Lie algebras.
We prove Ptolemaean Inequality and Ptolemaeus' Theorem in the closure complex hyperbolic plane endowed with the Cygan metric.
Characterizes closures of mapping class group orbits on non-orientable surfaces.
problem Understanding closures of orbits in Teichmüller spaces for non-orientable surfaces.
method Analyzes closures in ML and PML for measured laminations, projective measured laminations, and points. result Characterizes closures of weighted two-sided curves in ML. Paper explores RKHS properties for derivative and integral operators.
problem Establishing sufficient conditions for reproducing property in RKHS.
method Establishing reproducing property for combinations of composition operators.
result Provides framework for regularized learning algorithms involving function values, gradients, or operators.
The article finds equivalence moves for links in specific manifolds using plat closure of braids.
problem Finding equivalence moves for links in Dunwoody and periodic Takahashi manifolds.
method Representing manifolds with Heegaard splitting and braids, determining equivalence moves algorithmically and computing them explicitly.
result Explicit computation of equivalence moves for some cases.
Simplified proof of foliation closure theorem for linear foliations.
problem Proving the closure of linear foliations on Riemannian manifolds.
method Direct geometric approach, focusing on projectable foliations and compatible connections.
result Smoothness of the closure of linear foliations directly proven.
We show that if a braid B can be parametrised in a certain way, then previous work can be extended to a construction of a polynomial f:R4→R2 with the closure of B as the link of an isolated singularity of f, showing that the closure of B is real algebraic. In particular, we prove that cl…
The paper develops a method to infer model parameters and shared dynamics from related physical systems using data.
problem Calibrating models to match data when detailed system properties and laws are unknown.
method Hierarchical Bayesian framework, adaptive surrogate models, bilevel optimization.
result Joint estimation of individual model parameters and shared dynamics using data from related systems.
Equivalent condition found for q-holonomic sequences.
problem Characterizing q-holonomic sequences.
method Elimination property for subsets of variables.
result Proved equivalence of q-holonomicity and elimination property.
We determine the condition on a given lens space having a realization as a closure of homology cobordism over a planar surface with a given number of boundary components. As a corollary, we see that every lens space is represented as a closure of homology cobordism over a planar surface with three boundary components. …
Normal closures of certain finite subgroups in automorphism and outer automorphism groups of free groups are determined.
problem Determining the normal closures of specific finite subgroups in automorphism and outer automorphism groups of free groups.
method Analyzing the structure of finite subgroups and their normal closures in Aut(Fn) and Out(Fn). result The normal closure of certain finite subgroups in Aut(Fn) and Out(Fn) are determined.