Paper solves equations of any length in a specific group.
problem Finding solutions to equations over torsion-free groups of arbitrary length.
method Examined three equations of arbitrary length and showed they have a solution under certain conditions.
result Equations have solutions if two relations among coefficients hold.
Paper shows solvability of certain equations over torsion-free groups.
problem Solvability of equations of arbitrary length over torsion-free groups.
method Analyzes equations containing no blocks of the form t−1git−1 and proves solvability under certain conditions. result Equations s(t)=1 have solutions over torsion-free groups if a single relation on coefficients holds. Study curves evolving on hypersurfaces with free boundaries, preserving length.
problem Evolution of curves on hypersurfaces with free boundaries.
method Nonlocal evolution equation with nonlinear boundary conditions, short-time existence, uniqueness, and parabolic energy estimates.
result Global existence and convergence to critical points proved.
Generalizes Hasimoto transformation to arbitrary flows on space curves.
problem Modeling and analyzing wave motions on space curves.
method Develops a mapping between curve evolution and scalar equations, transforming general vector fields in the Frenet frame.
result Binormal flows are generally length preserving but bending energy is fragile.
Hamilton's Ricci flow (RF) equations were recently expressed in terms of the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry, for d greater than or equal to 2. The structure of the simplicial Ricci flow (SRF) equations are dimensionally agnostic. These SRF equations were tested numerically and…
Geometric derivation of Einstein equations from causal fermion systems.
problem Deriving Einstein's equations from a new theoretical framework.
method Analysis of causal fermion systems and causal action principle.
result Einstein equations derived from causal action principle.
Paper trains a Transformer to add numbers of any length.
problem Training Transformers to handle arbitrary-length addition.
method Autoregressive generation from right to left.
result Trains a Transformer to generalize addition of numbers of any length.
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.
Rust library solves complex equations on abstract simplicial complexes.
problem Solving partial differential equations on abstract simplicial complexes.
method Finite Element Exterior Calculus, intrinsic Riemannian metric, first-order Whitney basis functions.
result Verification through convergence studies on elliptic Hodge-Laplace eigenvalue and source problems.
Edge subdivision affects the Perron eigenvalue of tree Ricci matrices.
problem Understanding how edge subdivision impacts the Perron eigenvalue of tree Ricci matrices.
method Compressing branches into scalar feedback functions via Schur complement, reducing the spectral problem to a one-dimensional Chebyshev equation.
result Edge subdivision can decrease, preserve, or increase the Perron eigenvalue of tree Ricci matrices.
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.
Bayesian methods detect clusters in noisy data more reliably.
problem Noisy data distorts traditional clustering methods, leading to unreliable results.
method Bayesian community detection using Minimum Description Length principle.
result Bayesian methods identify more robust clusters in noisy data.
I-BERT extends Transformer's self-attention to arbitrary input lengths.
problem Transformer models struggle with inductive generalization to unseen input lengths.
method Replaces positional encodings with a recurrent layer.
result I-BERT achieves state-of-the-art results on algorithmic tasks.
We present a separation property for the gaps in the length spectrum of a compact Riemannian manifold with negative curvature. In arbitrary small neighborhoods of the metric for some suitable topology, we show that there are negatively curved metrics with a length spectrum exponentially separated from below. This prope…
The study counts curves on a once-punctured torus with self-intersections.
problem Counting closed curves with self-intersections on a once-punctured torus.
method Combinatorial classification of curves with given word-length and self-intersections.
result Determination of curve counts with zero, one, and arbitrary self-intersections.
Formula for integrating random variables on hyperbolic surfaces.
problem Integrating random variables on the moduli space of hyperbolic surfaces.
method Integration formula for lengths of closed geodesics.
result Integral of geometric random variables can be expressed as an integral over R.
For word-equations in groups, we find a logarithmic bound on non-solutions.
problem Finding the length of non-solutions to word-equations in groups.
method Analyzing finite-rank free groups and applying results to broader classes of groups.
result Logarithmic bound on non-solutions for word-equations in groups.
This paper constructs PH spline curves with prescribed arc lengths.
problem Interpolating points, tangent directions, and curvatures with prescribed arc-length.
method Local construction of G2 planar PH biarc curves of degree 7. result Prescribed arc-length can be satisfied for any data and any chosen ratio between boundary tangents.
The study connects translation length to manifold structure, proving bounds and identifying finite types.
problem Understanding the structure of 3-manifolds via pseudo-Anosov mapping classes and their translation lengths.
method Proving bounds on translation lengths and constructing specific 3-manifolds from mapping tori.
result Finite set of 3-manifolds can be derived from pseudo-Anosov mapping classes with bounded translation length.
String vertices proven in hyperbolic geometry, unique up to transformations.
problem Existence and uniqueness of string vertices in string field theory.
method Homological proof using hyperbolic metrics and geodesic boundaries.
result String vertices are sets of surfaces with systole greater than or equal to L. Proves existence of curves with constant curvature in a sphere.
problem Existence of curves with constant geodesic curvature in a Riemannian 2-sphere.
method Develops a min-max scheme for a weighted length functional.
result Proves existence for almost every prescribed curvature.
This paper studies spectral properties of spheres with one equator.
problem Spectral rigidity and flexibility of spheres with one equator.
method Defined marked length spectrum, proved isospectrality, and classified contact forms.
result Marked length spectrum determines the metric up to Z2-symmetry. We solve linear equations with tensors of any rank.
problem Solving linear equations involving tensors of arbitrary rank.
method Developed a systematic approach for tensors of rank 3 and generalized to arbitrary rank.
result Derived a solution for tensors of arbitrary rank.
We construct a counterexample to a conjectured inequality L<2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2-sphere. The construction relies on Guillemin's theorem concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd def…
Alternative construction of quasi-Fuchsian flows using vortex equations.
problem Constructing quasi-Fuchsian flows.
method Using coupled vortex equations to construct quasi-Fuchsian flows as thermostats.
result Formulas for the marked length spectrum of quasi-Fuchsian flows.
An arbitrary homomorphism between groups is nonincreasing for stable commutator length, and there are infinitely many (injective) homomorphisms between free groups which strictly decrease the stable commutator length of some elements. However, we show in this paper that a random homomorphism between free groups is almo…
The paper develops formulas for hyperbolic simplices based on edge lengths.
problem Understanding the geometry of hyperbolic simplices using only edge lengths.
method Develops geometric formulas for hyperbolic simplices based on edge lengths.
result Distance and projection formulas in hyperbolic simplices.
A well known Conjecture due to Beloshapka asserts that all totally nondegenerate polynomial models with the length l≥3 of their Levi-Tanaka algebra are {\em rigid}, that is, any point preserving automorphism of them is completely determined by the restriction of its differential at the fixed point onto the comple…
Deep architecture such as hierarchical semi-Markov models is an important class of models for nested sequential data. Current exact inference schemes either cost cubic time in sequence length, or exponential time in model depth. These costs are prohibitive for large-scale problems with arbitrary length and depth. In th…
The quadric ansatz solves dKP equations in arbitrary dimensions, leading to Einstein-Weyl structures.
problem Characterizing solutions of the dispersionless KP equation in arbitrary dimensions.
method Quadric ansatz for the dKP equation, constructing Einstein-Weyl spaces.
result Explicit new family of Einstein-Weyl spaces constructed and characterized.
The study finds solutions to a curvature minimisation problem in fixed-length curves.
problem Minimizing the L∞-norm of curvature among curves of fixed length. method Characterized solutions by a system of differential equations and classified the structure of solutions.
result Characterized solutions to the L∞-norm of curvature problem. In an appendix to an earlier paper (cf. arXiv:1703.00984) we showed we showed how to construct tunnels of positive scalar curvature and of arbitrarily small length and volume connecting points in a \emph{three dimensional} manifold of \emph{constant sectional curvature}. Here we generalize the construction to arbitrary…
An online framework optimizes efficiency in conformal prediction with a target miscoverage rate.
problem Achieving coverage and minimizing interval length in a sequential, online setting.
method Optimizes efficiency by directly optimizing the average length of intervals while maintaining coverage.
result Shows a gap between optimal performance for exchangeable and arbitrary sequences, and provides a matching algorithm for the Pareto-optimal settings.
The paper connects Riemann surface length spectra to Brownian loop measures.
problem Understanding the length spectra of Riemann surfaces with additional cusps.
method Using the Brownian loop measure to relate length spectra of Riemann surfaces with and without additional cusps.
result Expressed the total mass of Brownian loops in terms of the length of geodesic representatives.
The paper solves pentagon equations using triangulations and edge transformations.
problem Solving pentagon equations with triangulations and edge transformations.
method General data and transformation rule method applied to triangulations.
result Recovery of initial data after transformations.
Here we develop some basic analytic tools to study compactness properties of J-curves (i.e. pseudo-holomorphic curves) when regarded as submanifolds. Incorporating techniques from the theory of minimal surfaces, we derive an inhomogeneous mean curvature equation for such curves, we establish an extrinsic monotonicity…
The study examines the asymptotic behavior of extremal length in Teichmüller space.
problem Understanding the asymptotic behavior of extremal length along Teichmüller rays.
method Analyzing the limit of extremal length and deriving formulas for limiting Teichmüller distance and detour metric.
result An explicit formula for the limiting Teichmüller distance and a necessary and sufficient condition for Teichmüller rays to be asymptotic.
Study on finiteness property of right-angled Artin groups actions on extension graphs.
problem Finiteness property of hyperbolic simplicial actions on right-angled Artin groups.
method Analysis of right-angled Artin group actions on extension graphs, using asymptotic translation lengths and syllable lengths.
result Asymptotic translation lengths of elements in right-angled Artin groups are rational and have a common denominator under certain conditions.
This paper discovers new identities linking geodesic and orthogeodesic lengths on hyperbolic surfaces.
problem Understanding relationships between geodesic and orthogeodesic lengths on hyperbolic surfaces.
method Investigates a broad family of identities involving lengths of all closed geodesics and orthogeodesics.
result Introduces new identities that include lengths of all closed geodesics, contrasting with previous identities.
New symmetries found for scalar and vector ODEs of arbitrary dimensions.
problem Identifying symmetries for scalar and vector ODEs of arbitrary dimensions.
method Explicit expressions and abelian Lie algebra for non-Cartan symmetries in arbitrary dimensions.
result Non-Cartan symmetries characterize linearizable systems of ODEs but not nonlinear ones.
The Fu-Yau equation is an equation introduced by J. Fu and S.T. Yau as a generalization to arbitrary dimensions of an ansatz for the Strominger system. As in the Strominger system, it depends on a slope parameter α′. The equation was solved in dimension 2 by Fu and Yau in two successive papers for α′>0, and for $…
Mirzakhani extended curve counting from simple to all curves.
problem Counting curves in the mapping class group orbit with length at most L.
method Low-tech argument showing general result from simple curves.
result Derived the asymptotic growth of curves for arbitrary curves.
A pseudo-length function defined on an arbitrary group G=(G,⋅,e,()−1) is a map ℓ:G→[0,+∞) obeying ℓ(e)=0, the symmetry property ℓ(x−1)=ℓ(x), and the triangle inequality ℓ(xy)⩽ℓ(x)+ℓ(y) for all x,y∈G. We consider pseudo-length functions which sa…
We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope α′. Actually we obtain at the same time a solution of the open case α′>0, an improved solution of the known case α′<0, and solutions for a family of Hessian equations which includes the Fu-Yau equation as a special case. The method is based …
Here a new notion of fractional length of a smooth curve, which depends on a parameter σ, is introduced that is analogous to the fractional perimeter functional of sets that has been studied in recent years. It is shown that in an appropriate limit the fractional length converges to the traditional notion of length u…
Let α(s) be an arc on a connected oriented surface S in E3, parameterized by arc length s, with torsion τ and length l. The total square torsion F of α is defined by T=\int_{0}^{l}τ^{2}ds\ $. . The arc α is called a relaxed elastic line of second kind if it is an extremal for the variational problem of minimizing the v…
Let α be an arc on a connected oriented surface S in Minkowski 3-space, parameterized by arc length s, with torsion τ and length l. The total square torsion H of α is defined by . The arc is called a relaxed elastic line of second kind if it is an extremal for the variational prob…
Study Euler characteristics and loop lengths in hyperbolic 3-manifolds.
problem Analyzing Euler characteristics and loop lengths in hyperbolic 3-manifolds.
method Examining subgroups generated by loops and their index of freedom.
result Euler characteristic is bounded by the index of freedom.