Randomized positional encodings boost transformer performance on longer sequences.
problem Transformers struggle with generalizing to sequences of arbitrary length.
method Introduced randomized positional encodings that simulate longer sequences and randomly select positions.
result Randomized positional encodings increase test accuracy by 12.0% on average for sequences of unseen length.
BiPE blends intra-segment and inter-segment encodings for better length extrapolation.
problem Improving length extrapolation in language models.
method Bilevel Positional Encoding (BiPE) that separates intra-segment and inter-segment encodings.
result BiPE enhances length extrapolation across various text modalities.
The study shows that certain manifolds cannot have metrics with positive scalar curvature.
problem The existence of metrics with positive scalar curvature on manifolds.
method Definition and analysis of enlargeable length-structures and their properties.
result Closed manifolds with certain properties are obstructions to the existence of metrics with positive scalar curvature.
New curves generalize flat metrics from quadratic to q-differentials.
problem Determining flat metrics from curve lengths.
method Introduced q-simple curves to generalize results from quadratic to q-differentials.
result Lengths of q-simple curves uniquely determine non-positively curved Euclidean cone metrics induced by q-differentials.
Study approximate marked length spectrum rigidity in non-positively curved groups.
problem Approximate rigidity of marked length spectra in non-positively curved groups.
method Compare marked length spectra of isometric actions of groups with non-positively curved features.
result Supremum of quotient of marked length spectra is approximately determined by restricted spectra.
This paper establishes the existence of a gap for the stable length spectrum on a hyperbolic manifold. If M is a hyperbolic n-manifold, for every positive e there is a positive d depending only on n and on e such that an element of pi_1(M) with stable commutator length less than d is represented by a geodesic with leng…
3D manifold with positive curvature has a short geodesic.
problem Finding the shortest closed geodesic in 3-manifolds with positive scalar curvature.
method Analyzing the properties of geodesics in manifolds with positive scalar curvature.
result Existence of a non-trivial closed geodesic of length less than 22500.
New projection complex shows some surface homeomorphisms have positive commutator length.
problem Understanding commutator length in surface homeomorphisms.
method Constructing unbounded quasi-trees and a new projection complex.
result Some surface homeomorphisms have positive stable commutator length.
The paper proves critical lengths for loops on positively curved manifolds.
problem Existence and properties of closed geodesics on positively curved manifolds.
method Analyzing critical lengths of loops and geodesics on Riemannian manifolds with positive sectional curvature.
result Critical lengths attain their maximal value of 2π only for the round metric on the n-sphere.
Study finds infinite closed geodesics on all compact 2-orbifolds.
problem Existence of closed geodesics on compact 2-orbifolds.
method Analyzes properties of Riemannian 2-orbifolds.
result Infinitely many closed geodesics of positive length on compact 2-orbifolds.
Decomposes geodesic currents on surfaces into measured laminations or positive systole components.
problem Decomposing geodesic currents on surfaces of finite type.
method Topological decomposition and analysis of intersection functions.
result Currents with positive systole are bilipschitz equivalent to length functions under hyperbolic metrics.
Geometric characterization of sub-Riemannian geodesics on frame bundles.
problem Characterize sub-Riemannian geodesics on frame bundles of 3-manifolds.
method Lie theoretical description, geometric characterization, complex length spectrum computation.
result Sub-Riemannian metrics on frame bundles of isospectral manifolds are length isospectral.
Anosov maps on torus curve graphs have positive integer translation lengths.
problem Understanding the translation lengths of Anosov maps on curve graphs of tori.
method Constructive proof and algorithm for calculating exact translation lengths.
result The stable translation length of an Anosov map on the curve graph is always a positive integer.
Describes geometry of positive configurations in limiting buildings.
problem Understanding positive representations and their limits in buildings.
method Uses positivity properties of Hitchin representations and Parreau's compactification.
result Explicitly describes the geometry of preferred apartments in limiting buildings.
Derive derivatives of length functions on Teichmüller spaces using shearing coordinates.
problem Compute derivatives of length functions on Teichmüller spaces.
method Use shearing coordinates and Bonahon's theory of transverse H{ö}lder distributions.
result Hessian of length functions is positive-definite if curves intersect every leaf of a maximal geodesic lamination.
Transformer struggles with arithmetic length but improves with explicit structure encoding.
problem Transformers fail to generalize length in arithmetic tasks.
method Explicitly encoding structural symmetries via modified number formatting and custom positional encodings.
result Transformer can generalize up to 50-digit numbers without additional data.
The (1,1)-length of a knot is a braid group invariant equaling its level number.
problem Finding a numerical invariant for knot positions.
method Using braid group on two points in the torus to describe and calculate the (1,1)-length. result The (1,1)-length equals the level number of a knot. Dominant representations found via Fock-Goncharov coordinates.
problem Finding dominant representations of surface groups.
method Linear-algebraic approach using Fock-Goncharov coordinates.
result Explicit description of dominating representations.
Construct tunnels of positive scalar curvature in arbitrary dimensions.
problem Construct tunnels connecting points in manifolds of constant sectional curvature.
method Generalized construction to arbitrary dimensions, requiring only scalar curvature positivity.
result Existence of arbitrarily narrow tunnels of prescribed length.
The paper provides uniform length estimates for trajectories on flat cone surfaces.
problem Estimating the length of trajectories on flat cone surfaces.
method Using self-intersection numbers and constants depending only on the flat metric, the paper focuses on convex flat cone spheres with a positive curvature gap and a fixed number of singularities.
result Uniform two-sided estimates for trajectory lengths on convex flat cone spheres are obtained.
Proposes a differentiable STFT for more efficient optimization of hop length.
problem Efficient optimization of hop length in STFT for better temporal control.
method Introduces a differentiable version of STFT with continuous hop length.
result Improves optimization methods like gradient descent for STFT.
We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.
We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.
Counterexample disproves Matsumoto's conjecture on Finsler manifolds.
problem Matsumoto's conjecture on absolute vs. relative lengths in Finsler manifolds.
method Provided a counterexample to disprove the conjecture.
result Matsumoto's conjecture is false for certain Finsler manifolds.
The study examines conjugation curvature in a specific group, finding elements with various curvatures.
problem Analyzing conjugation curvature in a particular group structure.
method Examined elements in BS(1,n), calculated word length, and used density results. result Found elements with positive, negative, and zero conjugation curvature.
The study examines arithmetic orbifolds and their length spectra, proving uniform discreteness and linear dependence of geodesic lengths.
problem Uniform discreteness and linear dependence of geodesic lengths in arithmetic orbifolds.
method Analyzes Salem numbers and Lie groups to prove uniform discreteness, and uses geometric properties to show linear dependence of geodesic lengths.
result Existence of a positive constant δ(X) such that squares of lengths of closed geodesics shorter than δ must be pairwise linearly dependent over Q.
We establish bounds on the minimal asymptotic pseudo-Anosov translation lengths on the complex of curves of orientable surfaces. In particular, for a closed surface with genus g⩾2, we show that there are positive constants a1<a2 such that the minimal translation length is bounded below and above by $a…
Let Γ be a finite index subgroup of the mapping class group MCG(Σ) of a closed orientable surface Σ, possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element g∈Γ has positive stable commutator length. In addition, we show that in these situations th…
Study of manifolds with prime cyclic group actions and curvature properties.
problem Curvature properties of manifolds with Zp-actions. method Analysis of Zpr-actions on positively curved manifolds, use of error-correcting codes. result Improved symmetry-rank bounds for n-manifolds with p-actions, especially for small primes. For non-reversible Finsler metrics of positive flag curvature on spheres and projective spaces we present results about the number and the length of closed geodesics and about their stability properties.
For hyperbolic surfaces, primitive lengths are bounded below by a specific formula.
problem Understanding the distribution of primitive closed-geodesic lengths on hyperbolic surfaces.
method Analyzing Teichmüller space and proving a lower bound on the number of distinct primitive lengths.
result There exists a lower bound on the number of distinct primitive closed-geodesic lengths for hyperbolic surfaces.
Constructs symplectic surface bundles with positive signatures.
problem Symplectic surface bundles over surfaces with positive signatures.
method Constructs symplectic surface bundles with specific genera.
result Determines commutator lengths of new mapping classes.
Characterizes paths minimizing anisotropic lengths in Euclidean space.
problem Finding paths of minimal anisotropic length between points.
method Characterization through geometric connection to anisotropic isoperimetric set.
result Established a connection between minimizing paths and anisotropic isoperimetric geometry.
Splitting theorem for non-positively curved Lorentzian spaces.
problem Understanding curvature in Lorentzian spaces.
method Proving a splitting theorem with global non-positive timelike curvature and extending first variation formula.
result Splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature.
The paper generalizes rectifying and normal curves in Lorentzian n-space.
problem Characterizing and classifying g−rectifying and g−normal curves in Lorentzian n-space. method Introducing a g−position vector field and defining g−rectifying and g−normal curves based on this field. result Comprehensive characterization and classification of g−rectifying and g−normal curves. Existence of closed geodesics proven on specific orbifolds.
problem Proving the existence of closed geodesics on compact developable orbifolds.
method Stratification of singular locus, reduction to even-dimensional orbifolds with specific properties.
result Existence of closed geodesics of positive length on compact developable orbifolds of dimensions 3, 5, or 7.
We prove that all elements of infinite order in Out(Fn) have positive translation lengths; moreover, they are bounded away from zero. Consequences include a new proof that solvable subgroups of Out(Fn) are finitely generated and virtually abelian and the new result that such subgroups are quasi-convex.
We study metric and analytic properties of generalized lemniscates E_t(f)={z:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E_t(f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E_t(f)| is convex on any interval free of cri…
GIST adapts HMC by tuning parameters based on position and momentum.
problem Locally adaptive sampling in Hamiltonian Monte Carlo.
method GIST uses Gibbs sampling to adaptively tune HMC parameters.
result GIST improves sampling efficiency for high-dimensional models.
Shortest geodesic on curved spheres is no longer than 3 times the diameter.
problem Finding the shortest closed geodesic on spheres with positive curvature.
method Proved a new isoperimetric inequality for spheres with pinched curvature, used to improve the bound on the shortest geodesic.
result The shortest closed geodesic is no longer than 3 times the diameter of the sphere.
New homeomorphism found in Klein bottle group.
problem Understanding homeomorphisms of Klein bottle.
method Using recent results on commutator length.
result Existence of homeomorphism with positive stable commutator length.
The study finds infinite commensurability classes of hyperbolic manifolds with Salem number lengths.
problem Understanding commensurability classes of arithmetic hyperbolic manifolds.
method Analyzing Salem numbers and their relation to arithmetic hyperbolic manifolds.
result Infinitely many commensurability classes of arithmetic hyperbolic manifolds with geodesic lengths equal to logarithms of Salem numbers.
We bound the dimension of the fiber of a Riemannian submersion from a positively curved manifold in terms of the dimension of the base of the submersion and either its conjugate radius or the length of its shortest closed geodesic.
Study bounds the length of shortest periodic geodesics on certain curved spaces.
problem Bounding the length of shortest periodic geodesics on curved spaces.
method Analyzing the space of closed loops and their homotopy.
result The length of a shortest periodic geodesic is bounded by 8π(n−1). We verify here some variants of topological and dynamical flavor of the injectivity radius conjecture in Hofer geometry, Lalonde-Savelyev \cite{citeLalondeSavelyevOntheinjectivityradiusinHofergeometry} in the case of Ham(S2) and Ham(Σ,ω), for Σ a closed positive genus surface. In particular we show that any lo…
New boundary for infinite surfaces defined using length spectrum.
problem Defining Thurston's boundary for infinite surfaces.
method Using length spectrum of infinite geodesic pairs of pants.
result Thurston's boundary is strictly larger than projective bounded measured laminations when lengths converge to zero.
New stability estimate for Anosov manifolds' metrics.
problem Locally determine the metric from the marked length spectrum.
method Anosov geodesic flow and non-positive curvature.
result Two close enough metrics with the same marked length spectrum are isometric.
We give a lower bound for the length of a non-trivial geodesic loop on a simply-connected and compact manifold of even dimension with a non-reversible Finsler metric of positive flag curvature. Harris and Paternain use this estimate in their recent paper [HP] to give a geometric characterization of dynamically convex F…