The study calculates the growth rate of reciprocal hyperbolic elements in Hecke groups.
problem Counting reciprocal hyperbolic elements in Hecke groups.
method Analyzes conjugacy classes of hyperbolic elements associated with reciprocal geodesics.
result Determines the asymptotic growth rate and limiting constant of primitive conjugacy classes of reciprocal hyperbolic elements.
Classifies reciprocal elements in Hecke groups, generalizing Sarnak's work.
problem Classifying reciprocal elements in Hecke groups.
method Classifying and parametrizing reciprocal classes in Hecke groups Γp for p≥3. result Generalizes Sarnak's result on reciprocal elements in the modular group.
Estimates growth of reciprocal classes in Hecke groups.
problem Estimating the growth of reciprocal conjugacy classes in Hecke groups.
method Using free product structure and word lengths of reciprocal elements, with tools from basic probability theory.
result Estimates the asymptotic growth of reciprocal conjugacy classes in Hecke groups.
The study examines the growth of reciprocal classes in Hecke groups, proving an asymptotic formula.
problem Analyzing the growth of reciprocal classes in Hecke groups.
method Utilizes the free product structure of Hecke groups, combinatorial counting, and recurrence relations.
result Proves an asymptotic formula for the number of reciprocal classes in Hecke groups.
The paper classifies reversible elements in Seifert-fibered spaces and braid groups.
problem Classifying reversible elements in Seifert-fibered spaces and braid groups.
method Classification of reversible elements in Fuchsian groups, application to Seifert-fibered groups, and analysis of 3-torsion elements.
result Classification and analysis of reversible and 3-torsion elements in Seifert-fibered spaces and braid groups.
Innovative ball bearing converts rotary to reciprocating motion.
problem Designing efficient motion conversion mechanisms.
method Closed curve envelopment theory and diameter-stroke ratio concept.
result Compact and vibration-reduced ball bearing design.
We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper CAT(0) spaces, elements acting hyperbolically on the Bass-Serr…
We rigorously define the Liouville action functional for finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that the classical action - the critical point of the Liouville action functional, considered as a funct…
Study shows Morse elements are common in acylindrically hyperbolic groups.
problem Understanding generic elements in acylindrically hyperbolic groups.
method Analyzing Morse elements and outer automorphisms.
result Morse elements are common in acylindrically hyperbolic groups.
We survey the renormalized volume of hyperbolic 3-manifolds, as a tool for Teichmuller theory, using simple differential geometry arguments to recover results sometimes first achieved by other means. One such application is McMullen's quasifuchsian (or more generally Kleinian) reciprocity, for which different arguments…
New examples of hyperbolic links with generalized torsion elements found.
problem Finding generalized torsion elements in the fundamental groups of hyperbolic links.
method Analyzing the Weeks manifold, figure-eight sister manifold, and Whitehead sister link to identify generalized torsion elements.
result First examples of hyperbolic links with link groups admitting generalized torsion elements.
The paper constructs hyperbolic elements in multiple spaces.
problem Constructing hyperbolic elements in multiple Gromov-hyperbolic spaces.
method Explicit construction under minimal conditions.
result Set of simultaneously hyperbolic elements has strictly positive density.
Reciprocal processes are acausal generalizations of Markov processes introduced by Bernstein in 1932. In the literature, a significant amount of attention has been focused on developing dynamical models for reciprocal processes. Recently, probabilistic graphical models for reciprocal processes have been provided. This …
The paper models social networks with varying levels of reciprocity.
problem Understanding diverse reciprocal behavior in social networks.
method Developed a preferential attachment model with heterogeneous reciprocity.
result Captures the heavy-tailed nature of empirical degree distributions and identifies multiple user groups.
The study shows pseudo-Anosovs are common in mapping class groups.
problem Counting pseudo-Anosovs in mapping class groups.
method Using weakly contracting isometries and Morse elements.
result Pseudo-Anosovs are generic in mapping class groups.
New group with non-loxodromic Morse element found.
problem Finding non-loxodromic Morse elements in groups.
method Small-cancellation techniques to construct a Morse local-to-global group.
result Found an infinite-order Morse element that is not loxodromic.
The study finds infinitely many hyperbolic 3-manifolds with large rank and generalized torsion elements.
problem Finding hyperbolic 3-manifold groups with large rank and generalized torsion elements.
method Constructing specific hyperbolic 3-manifolds with given properties.
result Infinitely many hyperbolic 3-manifolds with generalized torsion elements of arbitrarily large order.
Formulates Hilbert reciprocity law on 3-manifolds.
problem Developing arithmetic topology on 3-manifolds.
method Formulated an analogue of Hilbert reciprocity law using intersection forms and Kummer extensions.
result Cyclic covers of links are analogues of Kummer extensions.
In this paper, we establish that, for statistically convex-cocompact actions, contracting elements are exponentially generic in counting measure. Among others, the following exponential genericity results are obtained as corollaries for the set of hyperbolic elements in relatively hyperbolic groups, the set of rank-1 e…
Minimal spectral radii found for specific matrix types.
problem Finding smallest spectral radii for certain matrix classes.
method Analyzing skew-reciprocal integer matrices of fixed even dimensions.
result Most classes of matrices have smaller spectral radii than their reciprocal counterparts.
The heat coefficients related to the Laplace-Beltrami operator defined on the hyperbolic compact manifold $H^3/\Ga$ are evaluated in the case in which the discrete group $\Ga$ contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace…
New infinite family of hyperbolic L-space knots with specific semigroups.
problem Characterizing semigroups of L-space knots.
method Defined formal semigroups from Alexander polynomials and analyzed hyperbolic knots.
result Found an infinite family of hyperbolic L-space knots with semigroups generated by five elements.
Study on geodesics and dihedral groups in lattices.
problem Growth and distribution of conjugacy classes of dihedral subgroups.
method Generalizing earlier work on reciprocal geodesics, proving equidistribution.
result Reciprocal geodesics are equidistributed in the unit tangent bundle.
We introduce the notion of a "crystallographic sphere packing," defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit for the first time an infinite family of conformally-inequivalent such with all radii being reciprocals of integers. We then…
Reciprocal processes are acausal generalizations of Markov processes introduced by Bernstein in 1932. In the literature, a significant amount of attention has been focused on developing dynamical models for reciprocal processes. In this paper, we provide a probabilistic graphical model for reciprocal processes. This le…
The paper models reciprocity in interbank markets using a statistical null model.
problem Understanding the importance of individual banks in financial networks.
method Developed an exponential random graph model to account for reciprocal links on both topological and weighted levels.
result Weighted reciprocity in interbank markets is more significant than network size and volume before the financial crisis.
Reciprocal learning unifies various machine learning algorithms.
problem Understanding the convergence of machine learning algorithms.
method Introducing reciprocal learning as a generalization of various algorithms using decision theory.
result Reciprocal learning algorithms converge at linear rates to an approximately optimal model under certain conditions.
Study geodesics entering a fixed cusp neighborhood multiple times.
problem Understanding geodesics entering a specific cusp neighborhood multiple times.
method Investigate reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
result Characterized the class of reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
New method reconstructs interbank networks enforcing reciprocity to improve stability and risk prediction.
problem Lack of public interbank network data and difficulty in replicating cycles.
method Proposes a new network reconstruction method enforcing sparsity and link reciprocity from aggregate data.
result Adding reciprocity improves prediction of network properties, including largest real eigenvalue and eccentricity of eigenvalues.
Persistent elements are ubiquitous in knot groups, especially for hyperbolic knots.
problem Identifying persistent elements in knot groups under Dehn fillings.
method Combining techniques from knot theory and hyperbolic geometry, including Dehn fillings and automorphisms.
result Persistent elements are structurally pervasive in knot groups, not just rare exceptions.
We prove that the Hilbert geometry of a convex domain in Rn has bounded local geometry, i.e., for a given radius, all balls are bilipschitz to a euclidean domain of Rn. As a consequence, if the Hilbert geometry is also Gromov hyperbolic, then the bottom of its spectrum is strictly positive. We…
Growth rates of geodesics on modular orbifolds are studied.
problem Understanding growth rates of geodesics on modular orbifolds.
method Exhaustion of modular orbifold by compact subsurfaces, analysis of low lying geodesics and reciprocal geodesics.
result Growth rates of low lying geodesics and reciprocal geodesics converge to the full set's growth rate.
This paper studies the generic behavior of k-tuple elements for k≥2 in a proper group action with contracting elements, with applications towards relatively hyperbolic groups, CAT(0) groups and mapping class groups. For a class of statistically convex-cocompact action, we show that an exponential generic set of …
The paper proves generalization bounds and stopping rules for self-selected data in reciprocal learning.
problem Generalization of learning algorithms using self-selected data.
method Proves universal generalization bounds using covering numbers and Wasserstein ambiguity sets.
result Provides stopping rules for reciprocal learning algorithms to ensure out-of-sample performance.
We prove that if φ,ψ∈Out(FN) are hyperbolic iwips (irreducible with irreducible powers) such that <φ,ψ>≤Out(FN) is not virtually cyclic then some high powers of φ and ψ generate a free subgroup of rank two, all of whose nontrivial elements are again hyperbolic iwips. Being a hyperbolic iwip element of $…
Study shows Poisson boundary matches hyperbolic boundary for certain groups.
problem Identifying Poisson boundary for hyperbolic groups without moment conditions.
method Proved using finite entropy random walks and extended to groups with WPD elements.
result Poisson boundary matches hyperbolic boundary for specified groups.
This work improves KG embeddings by integrating hyperbolic and attention mechanisms.
problem Preserving hierarchical and logical patterns in KGs with low-dimensional embeddings.
method Combines hyperbolic reflections/rotations with attention mechanisms to capture complex relational patterns.
result Improves MRR by up to 6.1% on standard benchmarks and new state-of-the-art results in high dimensions.
We obtain an exact modularity relation for the q-Pochhammer symbol. Using this formula, we show that Zagier's modularity conjecture for a knot K essentially reduces to the arithmeticity conjecture for K. In particular, we show that Zagier's conjecture holds for hyperbolic knots K=72 with at most seven cros…
The abstract discusses conjectures about Chern-Simons invariants of 3-manifolds.
problem Conjectures about the reciprocity of Chern-Simons invariants of 3-manifolds.
method Supporting evidence through Galois descent of a K3-group. result The conjectures hold under the condition of Galois descent of a K3-group. Given a knot and an SL(n,C) representation of its group that is conjugate to its dual, the representation that replaces each matrix with its inverse-transpose, the associated twisted Reidemeister torsion is reciprocal. An example is given of a knot group and SL(3,Z) representation that is not conjugate to its dual for …
Central limit theorem for Green metrics on hyperbolic groups.
problem Proving a central limit theorem for Green metrics on hyperbolic groups.
method Proving a central limit theorem for Green metrics on hyperbolic groups using probability measures and ordering elements.
result Proved a central limit theorem for Green metrics on hyperbolic groups.
Let Sp(2,1) be the isometry group of the quaternionic hyperbolic plane HH2. An element g in Sp(2,1) is `hyperbolic' if it fixes exactly two points on the boundary of HH2. We classify pairs of hyperbolic elements in Sp(2,1) up to conjugation. A hyperbolic element of $S…
Reciprocity laws for line bundles on circle fibrations over complex manifolds.
problem Analytic reciprocity laws for complex line bundles on fibrations in oriented circles.
method Study of Gysin maps and first Chern classes in cohomology.
result Sum of Gysin maps of Chern classes equals zero in H3(B,Z) under specific conditions. We study the geometry of the foliation by constant Gaussian curvature surfaces (Σk)k of a hyperbolic end, and how it relates to the structures of its boundary at infinity and of its pleated boundary. First, we show that the Thurston and the Schwarzian parametrizations are the limits of two families of parametrizati…
We reformulate Lehmer's question from 1933 and a question due to Schinzel and Zassenhaus from 1965 in terms of a comparison of the Mahler measures and the houses, respectively, of monic integer reciprocal and skew-reciprocal polynomials of the same degree. This entails that understanding the difference between orientat…
We show that the characteristic series for the greedy normal form of a Coxeter group is always a rational series, and prove a reciprocity formula for this series when the group is right-angled and the nerve is Eulerian. As corollaries we obtain many of the known rationality and reciprocity results for the growth series…
Improved method for numerical conformal mappings on complex domains.
problem Accurate and efficient computation of conformal mappings on multiply connected domains.
method Generalization and refinement of the conjugate function method using high-order finite element methods.
result Achieved accurate and efficient construction of boundary values for multiply connected domains.
Study of Bowditch representations in hyperbolic spaces with implications for dynamics and recognition.
problem Characterizing and understanding representations of free groups into hyperbolic spaces.
method Generalization of Bowditch conditions, explicit constant Kδ for hyperbolicity, characterizations of representations. result Linear growth of lengths for primitive elements in Bowditch representations, new characterization of primitive-stable representations.