Researchers solved a problem about triangle groups and their cusps.
problem Determine when functions on triangle group cusps are congruence.
method Computed the exact closure of triangle groups inside the profinite completion.
result Obtained conditions for functions to be congruence.
Embeds Higson compactification into adelic solenoids.
problem Embedding Higson compactification into solenoids.
method Induces isomorphism of 1-dimensional cohomology.
result Higson compactification embeds into adelic solenoids.
In this paper we represent the classical braids in the Yokonuma--Hecke and the adelic Yokonuma--Hecke algebras. More precisely, we define the completion of the framed braid group and we introduce the adelic Yokonuma--Hecke algebras, in analogy to the p--adic framed braids and the p--adic Yokonuma--Hecke algebras in…
The p-adic theory of the stock market is presented. It is shown that the price dynamics is very naturally described by the adelic function. The procedure of derivation of the functional integral formulation of adelic type is derived from microscopic models using generalized supercoherent states.
Study continuity of phi-invariant for degenerating graphs.
problem Continuity of phi-invariant for degenerating graphs.
method Use Yuan--Zhang's adelic divisors and follow Yuan's globalization of phi-invariants.
result Asymptotic expression of Zhang--Kawazumi's invariants for Riemann surfaces near the boundary of the moduli space.
Given a pair of number fields with isomorphic rings of adeles, we construct bijections between objects associated to the pair. For instance we construct an isomorphism of Brauer groups that commutes with restriction. We additionally construct bijections between central simple algebras, maximal orders, various Galois co…
Study shows automorphisms of Markov surfaces share periodic points if they share a common iterate.
problem Study of unlikely intersections for automorphisms of Markov surfaces with positive entropy.
method Arithmetic equidistribution for adelic line bundles, theory of laminar currents, quasi-Fuchsian representation theory.
result Two automorphisms with positive entropy share a Zariski dense set of periodic points if and only if they share a common iterate.
The reduced norm-one group G of a central simple algebra is an inner form of the special linear group, and an involution on the algebra induces an automorphism of G. We study the action of such automorphisms in the cohomology of arithmetic subgroups of G. The main result is a precise formula for Lefschetz numbers of au…
We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings…
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.
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.
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.
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…
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.
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. 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…
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. Specialized knot theory theorems for strongly involutive links.
problem Classical Alexander and Markov theorems for links.
method Equivariant closure map for strongly involutive links.
result Surjective equivariant closure map up to equivalence of strongly involutive links.
New method for knot closures from 1-tangles and annulus twists.
problem Understanding knot closures from 1-tangles and annulus twists.
method Wrapping index, sutured manifold theory, band sums.
result At most two closures for a given nontrivial 1-tangle, and one twist producing an unknot if it exists.
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.
Study horocycle orbits in Z-covers of hyperbolic surfaces.
problem Classify horocycle orbit closures in Z-covers of compact hyperbolic surfaces. method Careful analysis of distance minimizing geodesic rays in the cover.
result All non-maximal horocycle orbit closures have integer Hausdorff dimension.
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.
Let n be a positive integer. We provide a Khovanov homology proof of the following classical fact: If the closure of an n-strand braid is the n-component unlink, then the braid is trivial.
We study the Morton-Franks-Williams inequality for closures of simple braids (also known as positive permutation braids). This allows to prove, in a simple way, that the set of simple braids is a orthonormal basis for the inner product of the Hecke algebra of the braid group defined by Kálmán, who first obtained this r…
Floating geodesic planes in Hitchin manifolds have fractal closures with non-integer dimensions.
problem Rigidity of geodesic planes in Hitchin manifolds.
method Constructing a specific surface group and analyzing its action on the Hitchin manifold.
result Existence of floating geodesic planes in Hitchin manifolds with fractal closures.
Researchers use Gaussian processes to approximate Lagrange multipliers for Maximum-Entropy distributions.
problem Finding Lagrange multipliers for Maximum-Entropy distributions is computationally challenging.
method Employed Gaussian processes to approximate the Lagrange multipliers as a map of moments. Optimized hyperparameters by maximizing log-likelihood.
result Data-driven Maximum-Entropy closure performs well in approximating non-equilibrium distributions.
Derivation of reduced order representations of dynamical systems requires the modeling of the truncated dynamics on the retained dynamics. In its most general form, this so-called closure model has to account for memory effects. In this work, we present a framework of operator inference to extract the governing dynamic…
It has often been taken as a working assumption that directed links in information networks are frequently formed by "short-cutting" a two-step path between the source and the destination -- a kind of implicit "link copying" analogous to the process of triadic closure in social networks. Despite the role of this assump…
Generalizes Delzant theorem for torus-equivariantly embedded toric hypersurfaces.
problem Conditions for nonsingularity of complex subtorus orbits in symplectic toric manifolds.
method Clarification of Delzant theorem conditions using polytopes.
result Generalization of Delzant theorem for torus-equivariantly embedded toric hypersurfaces.