The paper studies medial and conflict sets under continuous deformations in Riemannian manifolds.
problem Understanding the behavior of medial and conflict sets under continuous deformations in Riemannian manifolds.
method A new approach using Kuratowski convergence to express the deformation process as a continuous process.
result Main `medial axis inner semi-continuity' result proved useful in computing tangent cones of the medial axis.
In this paper, we consider the intensity surface of a 2D image, we study the evolution of the symmetry sets (and medial axes) of 1-parameter families of iso-intensity curves. This extends the investigation done on 1-parameter families of smooth plane curves (Bruce and Giblin, Giblin and Kimia, etc.) to the general case…
Examines medial axis in pseudo-Euclidean spaces.
problem No specific problem stated; focuses on new context.
method Follows Birbrair and Denkowski's approach.
result Feasibility of medial axis in pseudo-Euclidean spaces checked.
The paper extends Alexander invariant and medial quandle equivalence to links.
problem Understanding stronger invariants for links than Alexander modules.
method Extending Joyce's observation to multiple medial quandles and reduced Alexander modules.
result Medial quandles provide stronger invariants than reduced Alexander modules for links.
Medial quandles are represented using a heterogeneous affine structure. As a consequence, we obtain numerous structural properties, including enumeration of isomorphism classes of medial quandles up to 13 elements.
Algorithm finds finite fundamental bikei for virtual knots.
problem Determining which virtual knots have finite fundamental bikei.
method Implemented an algorithm to complete presentation matrices to operation tables.
result Computed fundamental bikei for all prime virtual knots with up to four crossings.
The Blum medial axis rigidity is studied in terms of cross ratios and differential geometry.
problem Examining rigidity properties of the Blum medial axis under diffeomorphisms.
method Using cross ratios from projective geometry and differential geometry.
result Cross ratios and differential geometry uniquely determine the angles between smooth sheets.
Constructs the medial quandle from link's peripheral structure.
problem Building the medial quandle of links.
method From reduced Alexander module's peripheral structure.
result Medial quandle constructed successfully.
We consider a generic configuration of regions, consisting of a collection of distinct compact regions {Ωi} in Rn+1 which may be either smooth regions disjoint from the others or regions which meet on their piecewise smooth boundaries Bi in a generic way. We introduce a skeletal linking …
Study of Hom quandles, focusing on structure and counting.
problem Characterizing the structure of Hom quandles.
method Analyzing quandles of homomorphisms into medial quandles, focusing on medial source quandles.
result Characterization of Hom quandles, especially when the target is 2-reductive.
A quandle orbit's orientation is problematic when reversed.
problem The natural orientation-reversal of quandle orbits is unsuitable for medial quandles.
method Defined the orientation-reversal of a quandle orbit by inverting translations, observed it's unsuitable for medial quandles.
result The natural orientation-reversal of quandle orbits is unsuitable for medial quandles.
Study robust utility maximization with uncertain endowments.
problem Optimal strategy under nondominated model uncertainty.
method General representation result, Choquet's capacitability theorem, medial limits.
result Existence of optimal strategy and dual representation for optimal utility.
The paper extends a result about involutory medial quandles to links, providing bounds and stronger invariants.
problem Extending the involutory medial quandle concept to links and understanding its properties.
method Analyzing the involutory medial quandle of classical links and comparing it to the homology group.
result The order of the involutory medial quandle is bounded and can be infinite only if the determinant is zero.
Medial quandles fail to distinguish certain links, highlighting their limitations.
problem Detecting causality in spacetime using quandles.
method Investigated medial quandles' ability to distinguish specific links and knots.
result Medial quandles fail to distinguish certain links, including the connected sum of two Hopf links from an infinite series of relevant three-component links.
The study classifies graphs on surfaces with positive curvature properties.
problem Classifying graphs on surfaces with specific curvature properties.
method Using medial graphs and classification techniques.
result Complete classification of graphs on surfaces with positive Forman curvature and corner curvature.
Study links using quandles and groups, proving key properties.
problem Understanding links through algebraic structures.
method Analyzing modules and quandles, focusing on metabelian quotients and medial quandles.
result Fundamental multivariate Alexander quandle is isomorphic to the metabelian quotient of the link group.
A quandle is a self-distributive algebraic structure that appears in quasi-group and knot theories. For each abelian group A and c \in A we define a quandle G(A, c) on \Z_3 \times A. These quandles are generalizations of a class of non-medial Latin quandles defined by V. M. Galkin so we call them Galkin quandles. Each …
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.
Study of generalized Legendrian racks and their GL-structures.
problem Characterizing and classifying generalized Legendrian racks.
method Algebraic analysis, classification of racks and GL-racks, tensor products.
result Classification of several infinite families of GL-racks.
A short, elementary proof is given of the result: The number of components of a link arising from a medial graph M(G) by resolving vertices is equal to the nullity of the mod-2 Laplacian matrix of the graph G.
The article establishes a polynomial for signed cyclic graphs and links it to checkerboard colorability.
problem Understanding graphical virtual links and their properties.
method Constructing virtual links from signed cyclic graphs, proving checkerboard colorability, and introducing a polynomial F[G].
result A virtual link is graphical if and only if it is checkerboard colorable.
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.
The goal of this thesis is to study the singularities of the exponential map of Riemannian and Finsler manifolds (a concept related to caustics and catastrophes), and the object known as the cut locus (aka ridge, medial axis or skeleton), to improve existing results about its structure, to look at it in new ways, and t…
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…
We study the superreplication of contingent claims under model uncertainty in discrete time. We show that optimal superreplicating strategies exist in a general measure-theoretic setting; moreover, we characterize the minimal superreplication price as the supremum over all continuous linear pricing functionals on a sui…
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.
Enhances quandle and biquandle colorings using biquandle structures.
problem Improving quandle and biquandle colorings.
method Investigates biquandle structures and biquandle homomorphisms.
result Describes biquandle structure of the Hom-biquandle and relates Hom-quandles to Hom-biquandles.
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.
Mark all vertices on a curve evolving under a family of curves obtained by intersecting a smooth surface M with the 1-parameter family of planes parallel to the tangent plane to M at a point p. Those vertices trace out a set, called the vertex set of M through p. We take p to be an isolated umbilic point on M and descr…
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.
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.
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…
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.
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.
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…
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…
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.