The paper finds the smallest order closed sublattice containing a given sublattice in vector lattices.
problem Identifying the smallest order closed sublattice containing a given sublattice in vector lattices.
method Analyzing the order closure and second order closure of sublattices.
result The smallest order closed sublattice containing a sublattice Y is often the second order closure of Y. 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.
Paper shows infinite index for normal closure of half-twist powers in mapping class group.
problem Infinite index of normal closure of half-twist powers in mapping class group.
method Reformulated Jones representation using Hecke algebras and W-graphs.
result Normal closure of mth power of half-twist has infinite index in mapping class group of punctured sphere.
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.
Deep learning approximates system moments from data.
problem Approximating moments of spatial probabilistic systems.
method Dynamic Boltzmann Distributions (DBDs) with deep Boltzmann machines (DBMs).
result Learned moment closures improve generalization over traditional methods.
Extracts compact non-Markovian closure models from data.
problem Deriving reduced order representations of dynamical systems with memory effects.
method Operator inference using sparse polynomial regression and neural networks.
result The extracted models are compact and non-Markovian, capturing underlying dynamics.
Single-colored ADO-3 invariant matches Links-Gould polynomial for 5-braid closures.
problem Matching ADO-3 invariant with Links-Gould polynomial for specific knot types.
method Proved for closures of 5-braids, conjectured for all knots and links.
result Single-colored ADO-3 invariant equals Links-Gould polynomial for 5-braid closures.
New tangles found for 21 ribbon knots with 10 or fewer crossings.
problem Characterizing ribbon knots through tangle forms.
method For each of 21 ribbon knots, found a 4-strand tangle that satisfies specific closure properties.
result Each of 21 ribbon knots can be represented by a tangle that meets the specified closure conditions.
The paper solves PDEs from matrices with orthogonal columns, linking them to Hessian metrics and symmetric spaces.
problem Solving third order PDEs for strictly convex smooth functions.
method Geometric methods using Hessian metrics and symmetric spaces.
result Explicit solutions and a family of non-generic solutions with applications in Poisson geometry and Kahler structures.
Algorithm calculates Seifert matrices for colored links.
problem Computing Seifert matrices for colored links.
method Developed an algorithm implemented in Clasper software.
result Computes Seifert matrices, potential function, and signatures.
We introduce the concept of morphism of pseudogroups generalizing the étalé morphisms of Haefliger. With our definition, any continuous foliated map induces a morphism between the corresponding holonomy pseudogroups. The main theorem states that any morphism between complete Riemannian pseudogroups is complete, has a c…
Study shows torsion normal generators in non-orientable surface mapping class groups.
problem Understanding normal generators of mapping class groups for non-orientable surfaces.
method Analyzes periodic elements and their normal closures in non-orientable surfaces.
result Provides conditions for normal generation and examples of elements.
We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as cons…
The study examines vector fields with integer singularities in 3D balls.
problem Characterizing the strong Lp-closure of vector fields with finitely many integer singularities. method Characterization and decomposition of vector fields with finitely many integer singularities.
result Decomposition theorem for elements in LZ1(B), revealing information about mass-minimizing currents. We establish the second part of Milnor's conjecture on the volume of simplexes in hyperbolic and spherical spaces. A characterization of the closure of the space of the angle Gram matrices of simplexes is also obtained.
The relationships between braid ordering and the geometry of its closure is studied. We prove that if an essential closed surface F in the complements of closed braid has relatively small genus with respect to the Dehornoy floor of the braid, F is circular-foliated in a sense of Birman-Menasco's Braid foliation the…
The first part of this paper is a survey on Teichmueller curves and Veech groups, with emphasis on the special case of origamis where much stronger tools for the investigation are available than in the general case. In the second part we study a particular configuration of origami curves in genus 3: A "base" curve is i…
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.
Recently, Cochran and Harvey defined torsion-free derived series of groups and proved an injectivity theorem on the associated torsion-free quotients. We show that there is a universal construction which extends such an injectivity theorem to an isomorphism theorem. Our result relates injectivity theorems to a certain …
New methods for faster ranking and link prediction using higher-order motifs.
problem Real-time ranking and link prediction in applications like web search.
method Higher-order ranking and link prediction methods based on closing higher-order network motifs.
result The methods are faster and more efficient than existing methods based on closing triangles.
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.
This paper develops a path-first theory using signatures and jump lifts for self-exiting processes.
problem Developing a universal coordinate system for various types of paths and processes.
method Using signatures, jump lifts, and expected signatures, the paper presents a geometricity framework with algebraic properties and obstructions.
result The framework links various mathematical concepts and offers four main contributions to understanding and modeling self-exiting processes.
Paper refines braidoid equivalence for spherical knotoids.
problem Equivalence of knotoid diagrams in spherical space.
method Refined L-equivalence of braidoid diagrams. result Equivalence theorem for multi-knotoid diagrams in S2. 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.
The paper studies representations of braid groups via curves and finds conditions for their Zariski closure and arithmeticity.
problem Representations of braid groups via specific families of Riemann surfaces.
method Consider families of Riemann surfaces defined by plane curves and study their monodromy representations into symplectic groups.
result Criterions for the Zariski closure of the image of the representation to be maximal and for the image to be an arithmetic lattice.
Proposes a new model for RANS simulations with uncertainty.
problem Uncertainty in Reynolds-averaged Navier-Stokes simulations.
method Data-driven closure model with aleatoric uncertainty, Bayesian formulation, sparse indirect data.
result Accurate probabilistic predictions, even in regions of model error.
Intelligence emerges from stabilizing invariant cycles in memory.
problem Understanding the nature of intelligence and its emergence.
method Structural-dynamical account rooted in a topological closure law: \(\partial^2=0\).
result Memory-amortized inference (MAI) mechanism that implements SbS \(
ightarrow\) CCUP.
Classifies components of k-differentials and their orbit closures.
problem Classifying components of strata of k-differentials and their orbit closures.
method Algebraic approach using multiscale compactification.
result Complete classification of components of strata of holomorphic and meromorphic k-differentials.
Characterizes stable differentials in compactified strata of abelian differentials.
problem Describing the closure of abelian differentials with specific types of zeros and poles.
method Explicit characterization, complex analytic proof, and flat geometric proof for smoothing the boundary.
result Global residue condition from dual graph order for boundary differentials.
Classifies points in quaternionic hyperbolic spaces up to congruence.
problem Classifying points in quaternionic hyperbolic spaces up to congruence.
method Introduces geometric invariants and distance formulas to classify points.
result Congruence classes are described by quaternionic Cartan's angular invariants and distances.
Motivated by applications to bond markets, we propose a multivariate framework for discrete time financial markets with proportional transaction costs and a countable infinite number of tradable assets. We show that the no-arbitrage of second kind property (NA2 in short), recently introduced by Rasonyi for finite-dimen…
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 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 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.
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.
New game introduces linking-unlinking strategy for two-component links.
problem Tackling the linking and unlinking of two-component links.
method Introducing and analyzing the Linking-Unlinking Game on various link shadows.
result Winning strategies for specific link shadows are presented.
We show that every surface in H^hyp(4) is either a Veech surface or a generic surface, i.e. its GL^+(2,R)-orbit is either a closed or a dense subset of H^hyp(4) . The proof develops new techniques applicable in general to the problem of classifying orbit closures, especially in low genus. Recent results of Eskin-Mirzak…
Algorithm confirms non-order-preserving braids, proving infinite family not order-preserving.
problem Determining if a braid is non-order-preserving.
method Algorithm that checks non-order-preserving property of braids.
result Infinite family of simple 3-braids are not order-preserving.
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.
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…
This paper studies the normal closure of braid groups in a torus.
problem Understanding the normal closure of braid groups in a torus.
method Combining results from Birman and Goldberg, the paper explores the geometric interpretation of the normal closure of full braid groups.
result The normal closure of Bn(D) in Bn(T) has a geometric description. 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.