In Garside groups, axes of Morse elements are strongly contracting.
problem Understanding the dynamics of Morse elements in Garside groups.
method Analyzing the Cayley graph of Garside groups modulo their center, using Garside generators.
result Morse elements act loxodromically on the additional length graph of Garside groups.
Study on knot behavior under twisting, linking winding numbers, and braid axes.
problem Comparing Seifert and slice genera of knots in twist families.
method Analyzing the winding numbers and wrapping numbers of knots and their twist families.
result Characterization of braid axes and conditions for infinitely many tight fibered knots.
Paper extends link Floer homology detection to almost braided links.
problem Detecting links that are nearly braided using link Floer homology.
method Inspired by nearly fibered knots, uses clasp-braids and topological characterisation.
result Khovanov homology detects Whitehead link and infinite families of links.
The paper diagnoses factor models using characteristic axes and zero-curve restrictions.
problem Tackles systematic sign reversals and overcorrections in factor model pricing errors.
method Extends cap-axis integral diagnostic to general characteristic axes, measuring pricing errors as bridge-alpha curves.
result Axis-level pricing errors are nearly orthogonal to maximum-Sharpe gains, showing systematic sign reversals and overcorrections.
The paper diagnoses factor-model pricing errors using characteristic axes and bridge-alpha curves.
problem Tackles systematic sign reversals and overcorrections in factor-model pricing errors.
method Extends cap-axis integral diagnostic to characteristic axes, measures pricing errors as bridge-alpha curves, and uses a predetermined characteristic order to generate zero-curve restrictions.
result Axis-level pricing errors are nearly orthogonal to maximum-Sharpe gains, showing significant sign reversals and overcorrections.
We define a homology HN for closed braids by applying Khovanov and Rozansky's matrix factorization construction with potential axN+1. Up to a grading shift, H0 is the HOMFLYPT homology defined in arXiv:math/0505056. We demonstrate that, for N≥1, HN is a $\mathbb{Z}_2\o…
Bayesian optimization finds best hyperparameters for deep learning models.
problem Finding optimal hyperparameters in high-dimensional space.
method Bayesian optimization, Gaussian processes, and adaptive experimentation.
result Ax, BoTorch, and GPyTorch provide a powerful and simple framework for hyperparameter optimization.
Method evaluates disentanglement in DLVMs, including those not aligned with latent axes.
problem Evaluate disentanglement in DLVMs, especially those not aligned with latent axes.
method Proposes a statistical method to discover generative factors of a dataset.
result Empirically demonstrates the advantage of the method on two datasets.
This paper diagnoses factor-model pricing errors using a new method.
problem Measuring pricing errors in factor models with general characteristic axes.
method Developed a method to measure factor-model pricing errors as bridge-alpha curves, using a predetermined characteristic order and prefix portfolios.
result Adding a counterpart factor flips the curve's sign on every axis, but only HML and CMA overcorrect enough to be rejected.
Paper proposes AXE loss for non-autoregressive machine translation, improving performance.
problem Challenges in training non-autoregressive models due to lack of autoregressive factors and cross entropy loss penalties.
method Proposes aligned cross entropy (AXE) loss function using a differentiable dynamic program for better word order alignment.
result AXE-based training improves performance on major WMT benchmarks and sets a new state of the art for non-autoregressive models.
New method flattens decision boundary by targeting shortcut-aligned axes in disentangled latent space.
problem Shortcut learning in neural networks, leading to poor out-of-distribution generalization.
method Injects targeted anisotropic noise to regularize classifier sensitivity along shortcut-aligned axes.
result Achieves state-of-the-art OOD performance without shortcut labels or conflicting samples.
The paper classifies hypersurfaces in Riemannian manifolds with constant inner product and torse-forming axes.
problem Understanding hypersurfaces in Riemannian manifolds with specific geometric properties.
method Analyzing hypersurfaces with constant inner product and torse-forming axes.
result Classification of hypersurfaces with torse-forming axes.
We study the Lipschitz metric on Outer Space and prove that fully irreducible elements of Out(F_n) act by hyperbolic isometries with axes which are strongly contracting. As a corollary, we prove that the axes of fully irreducible automorphisms in the Cayley graph of Out(F_n) are stable, meaning that a quasi-geodesic wi…
Self-focal points on ellipsoids of dimension 3 or higher are rare.
problem Existence of self-focal points on Riemannian manifolds of dimension 3 or higher.
method Analyzing geodesics and umbilic points on ellipsoids of various dimensions.
result Ellipsoids of dimension 3 or higher with at least 4 distinct axes have no self-focal points.
AXE evaluates explanations to avoid misleading Rashomon set model selection.
problem Evaluating explanations for Rashomon set models to avoid false selection.
method Proposed AXE method to evaluate explanation quality.
result AXE detects adversarial fairwashing with 100% success rate.
Adding supplementary axes improves neural network learnability and accuracy.
problem Overfitting and computational cost in deep neural networks.
method Analysis of a simple MLP model and comparison with and without supplementary information.
result Neural networks with supplementary axes show more robust and accurate training results.
New findings on Fuchsian groups and their axes.
problem Determining the commensurability class of Fuchsian groups.
method Analyzing geodesic intersections and using a new approach.
result The conjecture holds for almost all Fuchsian groups.
Study inverse curve shortening flow on hyperbolic plane, classifying solitons.
problem Understanding the behavior of curves in hyperbolic geometry under a specific flow.
method Classifying solitons with respect to vector fields and studying their properties.
result Parabolic solitons are graphs on the y-axis, conformal solitons on the x-axis.
Proposes σ-PCA to learn identifiable linear transformations without whitening.
problem Cannot identify axes with equal variances in PCA.
method Unified model for linear and nonlinear PCA, introducing a missing piece to eliminate rotational indeterminacy.
result Eliminates subspace rotational indeterminacy in PCA.
The Conclusive Theorem solves Finsleroid metrics in 3D.
problem Determining the dependence of Finsleroid metrics on the azimuthal angle in 3D.
method Deriving algebraic and differential equations to solve for the metric functions.
result Explicit dependence of characteristic functions on the angle θ is obtained.
Researchers propose and solve a class of pseudo-Finslerian metrics with angle-separation.
problem Characterizing and solving pseudo-Finslerian metrics with specific angle-separation conditions.
method Derived complete algebraic and differential equations, solved the set for angle-regular solutions.
result Found and described an angle-regular solution for the Finsleroid-in-pseudo-Finsleroid type.
Classic braids embed in virtual braids.
problem Embedding classic braids in virtual braids.
method Elementary proof using generalizations of braid groups.
result Classical braid group injects into virtual braid group.
Recent results using inverse scattering techniques interpret every solution φ(x,y) of the sine-Gordon equation as a non-linear superposition of solutions along the axes x=0 and y=0. Here we provide a geometric method of integration, as well as a geometric interpretation. Specifically, every weakly regular surface…
Paper computes motion groups of links using TQFTs, proving a conjecture.
problem Computing representations of motion groups of links in S3. method Dimension reduction of Dijkgraaf-Witten (DW) TQFTs.
result Proves a conjecture for motion groups of closed manifolds and torus links.
This paper consists of two parts. In the first one we study the behaviour of medial axes (skeletons) of closed sets in a connected complete Riemannian manifold M under deformations. The second one is devoted to a similar study of conflict sets. We apply a new approach to the deformation process. Instead of …
The paper finds minimal generating sets and abelianizes the quasitoric braid group.
problem Understanding the structure of quasitoric braids and their subgroup properties.
method Provided two minimal generating sets and determined the abelianization.
result Minimal generating sets and abelianization of the quasitoric braid group were determined.
We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words…
Satellite links of fully positive braids are characterized.
problem Characterizing satellite links of fully positive braids.
method Analyzing fully positive braids and their satellites.
result Satellite links of fully positive braids are characterized by specific conditions.
Formula found for braid index of n-bridge braids.
problem Finding a formula for the braid index of n-bridge braids. method Elementary, effective, self-contained proof.
result Closed form formula for braid index of n-bridge braids. Most simple braids have positive topological entropy.
problem Understanding the topological entropy of simple braids.
method Reduction from simple braids to non-simple 3-strand braids.
result The proportion of simple braids with positive entropy approaches 100% as the number of strands increases.
This paper is concerned with detecting when a closed braid and its axis are 'mutually braided' in the sense of Rudolph. It deals with closed braids which are fibred links, the simplest case being closed braids which present the unknot. The geometric condition for mutual braiding refers to the existence of a close contr…
Paper proves Markov's theorem for extended welded braids and links.
problem Generalizing welded links to extended welded links and braids.
method Following Kamada's approach, proving Alexander and Markov's theorems for extended welded braids and links.
result Proves versions of Alexander and Markov's theorems for extended welded braids and links.
We show that 3-braid links with given (non-zero) Alexander or Jones polynomial are finitely many, and can be effectively determined. We classify among closed 3-braids strongly quasipositive and fibered ones, and show that 3-braid links have a unique incompressible Seifert surface. We also classify the positive braid wo…
Simpler proof for non-basic sets in 2D.
problem Proving non-basic sets in 2D.
method Defining Sternfeld arrays and proving non-basic sets.
result Simpler proof of non-basic sets in 2D.
The paper examines subgroup separability for surface and virtual braid groups.
problem Subgroup separability of surface and virtual braid groups.
method Study of subgroup separability (LERF) properties.
result Properties of subgroup separability for surface and virtual braid groups are explored.
Virtual braids are a combinatorial generalization of braids. We present abstract braids as equivalence classes of braid diagrams on a surface, joining two distinguished boundary components. They are identified up to isotopy, compatibility, stability and Reidemeister moves. We show that virtual braids are in a bijective…
On a Riemannian 2-torus (T2,g) we study the geodesic flow in the case of low complexity described by zero topological entropy. We show that this assumption implies a nearly integrable behavior. In our previous paper \cite{GK} we already obtained that the asymptotic direction and therefore also the rotation number ex…
Study on deformation cohomology for braided commutative structures.
problem Classifying and understanding deformations of braided commutative algebras.
method Extending Yang-Baxter Hochschild cohomology to braided commutative deformations.
result Classifies infinitesimal deformations of braided algebras that are braided commutative.
Polynomial-time algorithm for virtual braid triviality.
problem Determining if virtual braid words represent the trivial braid.
method An O(l3n)-time algorithm for VBn. result Determines triviality of virtual braid words efficiently.
This paper identifies braided 3-belts that can be written in a braid-only form.
problem Identifying braided 3-belts that can be written in a braid-only form.
method Developed an algorithm to calculate the braid word for braided 3-belts and determined the conditions for knotted boundaries.
result Identified the set of braided 3-belts that can be written in a braid-only form and derived a formula for the Jones polynomial for knotted boundaries.
Magic braids can be made with leatherworking technique.
problem Determine which braids can be made with leatherworking technique.
method Explore their relation to several braid groups.
result Magic braids can be made with leatherworking technique.
We consider the problem of estimating an unknown signal x0 from noisy linear observations y=Ax0+z∈Rm. In many practical instances, x0 has a certain structure that can be captured by a structure inducing convex function f(⋅). For example, ℓ1 norm can be used to encourage a sparse solution. T…
Study virtual braid groups, proving a key subgroup result.
problem Understanding congruence subgroups in virtual braid groups.
method Using an extension of the integral Burau representation.
result Proved level 2 congruence subgroup of virtual braid group is pure virtual braid group.
Machine learning classifies braids and discovers new invariants.
problem Classifying and discovering invariants of braids and flat braids.
method Supervised learning with neural networks to classify braids as trivial or non-trivial.
result Found new convenient invariants of braids, including a complete invariant of flat braids.
In the present paper we give a new method for converting virtual knots and links to virtual braids. Indeed the braiding method given in this paper is quite general, and applies to all the categories in which braiding can be accomplished. We give a unifying topological interpretation of virtuals and flats (virtual strin…
This paper extends braid lifting to coloured braid groupoids for all simple disc covers.
problem Lifting braids to homeomorphisms on branched covers of the disc.
method Defines a map from a coloured braid groupoid to a mapping class groupoid for all simple covers of the disc.
result Characterizes the lift of every coloured braid, recovering classical lifting on liftable braids.
Paper studies special braids from chromatic configuration spaces.
problem Understanding braids with crossing-changeable properties.
method Theory of chromatic configuration spaces and braids.
result Introduced braids with the ability to intersect and untangle.
We construct helicoid-like embedded minimal disks with axes along self-similar curves modeled on logarithmic spirals. The surfaces have a self-similarity inherited from the curves and the nature of the construction. Moreover, inside of a "logarithmic cone", the surfaces are embedded.