Study natural and conjugate mates of Frenet curves in Lie groups.
problem Characterize Frenet curves and their mates in Lie groups.
method Introduced natural and conjugate mates, derived relationships, analyzed specific curves.
result Obtained results for various Frenet curves in Lie groups.
New method proves mateability of triangle groups with Blaschke products.
problem Proving mateability of triangle groups with Blaschke products.
method Associating two piecewise analytic circle maps to the triangle group, mating these with Blaschke products, and constructing a common lift.
result Proves mateability of all cusped triangle groups with suitable Blaschke products.
Unified and generalized mating frameworks for Kleinian groups and rational maps.
problem Combining two frameworks for mating Kleinian groups with rational maps.
method Extended mating framework to genus zero hyperbolic orbifolds, constructed correspondences, defined parameter space.
result Explicit description and construction of conformal matings and correspondences.
This note gives sufficient conditions (isothermic or totally nonisothermic) for an immersion of a compact surface to have no Bonnet mate.
New framework mated Kleinian groups with complex polynomials, revealing unique group properties.
problem Mating Kleinian groups with complex polynomials dynamics.
method Orbit equivalence framework for holomorphic mating, focusing on Fuchsian groups and higher Bowen-Series maps.
result Only torsion-free Fuchsian groups can be mated, with specific properties of Bowen-Series maps.
The paper connects reflection groups to maps with specific dynamical properties.
problem Understanding the relationship between reflection groups and anti-rational maps.
method Established a correspondence using planar graphs.
result Complete answers to geometric mating problems for anti-rational maps.
In this work, we studied the properties of the spherical indicatrices of a Bertrand curve and its mate curve and presented some characteristic properties in the cases that Bertrand curve and its mate curve are slant helices, spherical indicatrices are slant helices and we also researched that whether the spherical indi…
We present a novel method for solving Canonical Correlation Analysis (CCA) in a sparse convex framework using a least squares approach. The presented method focuses on the scenario when one is interested in (or limited to) a primal representation for the first view while having a dual representation for the second view…
The Bonnet problem is solved for specific Thurston geometries.
problem Determining when an isometric immersion can be continuously deformed through isometric immersions preserving principal curvatures.
method Study of Bonnet pairs in Bianchi--Cartan--Vrănceanu spaces and Sol3; use of differential equations. result Uniqueness of Bonnet mates in specific Thurston geometries.
The paper studies degenerations of rational maps and their limits as geometrically finite rational maps.
problem Understanding the limits of quasi post-critically finite degenerations of rational maps.
method Constructing limits as geometrically finite rational maps on a tree of Riemann spheres, proving boundedness, and giving convergence criteria.
result Progress towards Thurston's compactness theorem and double limit theorem in complex dynamics.
Automated model tracks mouse behavior in home cages.
problem Confounding analysis in specialized arenas.
method Group Behaviour Model (GBM) using permutation matrix.
result Model captures individual and group behavior.
The study of Bonnet surfaces in 4D space forms reveals new conformally invariant properties and characterizes proper Bonnet surfaces.
problem Investigating Bonnet surfaces in 4D space forms with constant mean curvature.
method Analyzing the moduli space of congruence classes of isometric surfaces, studying properties of lines of curvature, and using infinitesimal isometric deformations.
result Isotropic isothermicity characterizes proper Bonnet surfaces and provides conditions for non-existence of Bonnet mates.
Overview of dynamics in algebraic correspondences and their connections.
problem Understanding dynamics in algebraic correspondences and their connections.
method Focus on matings between rational maps and Kleinian groups, highlighting unifying structures.
result Rich dynamics and connections between moduli spaces of rational maps and Kleinian groups.
In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the angle between the principal normal and an RM vector field for spherical curves. L…
With recent advances in high throughput technology, researchers often find themselves running a large number of hypothesis tests (thousands+) and esti- mating a large number of effect-sizes. Generally there is particular interest in those effects estimated to be most extreme. Unfortunately naive estimates of these effe…
We construct non-trapping asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points.
A condition for a statistical manifold to have an equiaffine structure is studied. The facts that dual flatness and conjugate symmetry of a statistical manifold are sufficient conditions for a statistical manifold to have an equiaffine structure were obtained in [2] and [3]. In this paper, a fact that a statistical man…
The paper tackles efficient computation of optimal transport by approximating conjugates with amortized optimization.
problem Efficient computation of convex conjugates in optimal transport is challenging and limits the quality of transport maps.
method The approach combines amortized approximations of conjugates with a fine-tuning solver to improve transport map quality.
result The method significantly improves the quality of transport maps for the Wasserstein-2 benchmark and models many 2D couplings and flows.
Estimates geodesics on surfaces without conjugate points.
problem Counting geodesics on surfaces without conjugate points.
method Margulis-type asymptotic estimates.
result Asymptotic estimates for geodesics on surfaces.
New method speeds up inference for non-conjugate Gaussian processes.
problem Inference for non-conjugate Gaussian processes is slow and unreliable.
method Automated augmented conjugate inference method that constructs auxiliary variables to make the model conditionally conjugate.
result Our method is up to two orders of magnitude faster and more robust than existing methods.
We introduce a multiple conjugation biquandle, and show that it is the universal algebra to define a semi-arc coloring invariant for handlebody-links. A multiple conjugation biquandle is a generalization of a multiple conjugation quandle. We extend the notion of n-parallel biquandle operations for any integer n, an…
The paper studies Jacobi fields and conjugate points in projective sprays.
problem Investigating Jacobi fields and conjugate points in projective sprays.
method Proved that conjugate points are preserved under projective changes and established conditions for the existence of conjugate points.
result Conditions for the existence of conjugate points in projectively deformed sprays.
Paper describes how to extend multiple conjugation quandles using maps.
problem Understanding affine extensions of multiple conjugation quandles.
method Introduces augmented MCQ Alexander pairs for affine extensions.
result Affine extensions of multiple conjugation quandles can be described by quadruples of maps.
For a surface group, a new bound is given for conjugator length function.
problem Finding an explicit bound for the conjugator length function of a surface group.
method Detailed analysis of conjugation reductions.
result An explicit bound n−1≤CL(2n)≤n+8g−1 for the conjugator length function of a surface group. The study examines Hopfian properties of conjugation quandles and their underlying groups.
problem Understanding the relationship between Hopfian properties of conjugation quandles and their underlying groups.
method Examined Hopfian and residual finiteness properties of conjugation quandles of specific groups.
result Conjugation quandles of Baumslag-Solitar groups are infinitely generated and not necessarily Hopfian.
Upper bound for conjugate radius in open manifolds with scalar curvature and spectrum constraints.
problem Bounding the conjugate radius of open manifolds with specific curvature and spectrum conditions.
method Established an upper bound using scalar curvature and bottom-of-spectrum constraints.
result For certain conditions, the conjugate radius is no more than π.
The questions when two Morse function on closed manifolds are conjugated is investigated. Using the handle decompositions of manifolds the condition of conjugation is formulated. For each Morse function on 3-manifold the ordered generalized Heegaard diagram is built. The criteria of Morse function conjugation are given…
Study conjugate locus in convex 3-manifolds using Jacobi fields.
problem Understanding conjugate points in convex 3-manifolds.
method Use Jacobi fields to define coordinate systems and classify singularities.
result Developed a novel method for determining conjugate points in 3D manifolds.
Paper finds new criteria for conjugate points in fluid flows.
problem Finding conjugate points in steady 2D Euler flows.
method Develops a new sufficient criterion for conjugate points, applies to any rotational cell, and uses a general construction of steady fluid surfaces.
result Improves on existing criteria and captures all known conjugate points in rotational cells.
Study analyzes Lévy process structure on manifolds with conjugate points.
problem Microlocal analysis of Lévy processes on manifolds with conjugate points.
method Microlocal analysis, pseudodifferential operators, Fourier integral operators.
result Generator can be expressed as sum of pseudodifferential and Fourier integral operators.
We study the geodesic X-ray transform X on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of X and relate it to the con…
In this paper, we establish upper bounds on the length of the shortest conjugator between pairs of infinite order elements in a wide class of groups. We obtain a general result which applies to all hierarchically hyperbolic groups, a class which includes mapping class groups, right-angled Artin groups, Burger--Mozes-ty…
Identifies conjugate points in spherical harmonics solutions of quasi-geostrophic equations.
problem Locating conjugate points in spherical harmonics solutions.
method Utilizing structure constants and quasi-geostrophic equations on the sphere, identifying conjugate points.
result Existence and location of conjugate points along spherical harmonics solutions.
Study of holomorphic correspondences combining entire maps and Fuchsian groups.
problem Understanding dynamics of entire maps and their interactions with Fuchsian groups.
method Systematic study of (∞:∞) holomorphic correspondences arising from conformal combinations of transcendental entire maps and Fuchsian groups. result The resulting correspondence is the composition of a Möbius involution and the deleted covering correspondence of a meromorphic function with a simple pole.
The paper calculates actions of string link operations for 4- and 5-component links.
problem Classifying link-homotopy classes of string links.
method Explicit calculation of actions of partial conjugations and conjugations for 4- and 5-component string links.
result Presentations of link-homotopy classes for 4- and 5-component links.
Study normal operators of double fibration transforms with conjugate points.
problem Normal operators of double fibration transforms with conjugate points.
method Stable conditions on the distribution of conjugate points, splitting into elliptic and Fourier integral operators.
result Normal operator splits into an elliptic pseudodifferential operator and Fourier integral operators.
Researchers compare different gradient methods for ridge regression, finding conjugate gradients have similar performance.
problem Comparing statistical properties of different gradient methods in ridge regression.
method Explicit non-standard error decomposition to bound prediction error of conjugate gradient iterates.
result Conjugate gradient iterates share optimality properties with gradient flow and ridge regression up to a constant factor.
The paper proves estimates and theorems for Kähler manifolds.
problem Curvature conditions on Kähler manifolds.
method Volume comparison and rigidity theorems.
result Conjugate radius estimates for Kähler manifolds.
The study shows conditions for thermostats to have no conjugate points.
problem Conditions for thermostats to have no conjugate points.
method Analyzes smooth functions and properties of thermostats.
result Proves conditions for thermostats to be projectively Anosov and have no conjugate points.
Alexander quandles can be embedded into groups.
problem Embedding Alexander quandles into groups.
method For any twisted conjugate quandle, find a group such that the quandle is embedded into the conjugation quandle of the group.
result Alexander quandles can be embedded into groups.
We derive and approximate the conjugate prior of Dirichlet and beta distributions.
problem Intractability of conjugate prior for Dirichlet and beta distributions.
method Derive conjugate prior, define closed-form approximation, and provide algorithm.
result Closed-form approximation enables fully tractable Bayesian treatment.
The paper defines Fenchel conjugate and biconjugate on Hadamard manifolds.
problem Defining Fenchel conjugate and biconjugate on curved spaces.
method Introduced a new definition of Fenchel conjugate and biconjugate on Hadamard manifolds based on the tangent bundle.
result Developed a Fenchel-Moreau Theorem for geodesically convex functions on Hadamard manifolds.
Geodesic flows on certain surfaces are shown to be semi-conjugate to expansive flows.
problem Understanding geodesic flows on compact surfaces without conjugate points.
method Time-preserving semi-conjugation to a continuous expansive flow.
result Geodesic flows on compact surfaces without conjugate points of genus > 1 have a unique measure of maximal entropy.
The paper studies conjugate points on Lie groups with specific metrics.
problem Existence and properties of conjugate points on Lie groups with left-invariant metrics.
method Using reformulated index form in terms of adjoint action, the paper proves sufficient conditions for conjugate points and provides bounds and criteria.
result All geodesics in compact semisimple Lie groups have conjugate points, with upper and lower bounds on conjugate times.
We determine the complete conjugate locus along all geodesics parallel or perpendicular to the center (Theorem 2.3). When the center is 1-dimensional we obtain formulas in all cases (Theorem 2.5), and when a certain operator is also diagonalizable these formulas become completely explicit (Corollary 2.7). These yield s…
We address the challenge of effective exploration while maintaining good performance in policy gradient methods. As a solution, we propose diverse exploration (DE) via conjugate policies. DE learns and deploys a set of conjugate policies which can be conveniently generated as a byproduct of conjugate gradient descent. …
In this paper, we study the Galois conjugates of stretch factors of pseudo-Anosov elements of the mapping class group of a surface. We show that - except in low-complexity cases - these conjugates are dense in the complex plane. For this, we use Penner's construction of pseudo-Anosov mapping classes. As a consequence, …
Study on conjugate points in a C∗-algebra's Grassmann manifold.
problem Characterizing conjugate points in a C∗-algebra's Grassmann manifold. method Analyzes connections and geodesics in the Riemannian metric induced by the Killing form.
result Points that are tangent conjugate in the classical setting may not be conjugate in a C∗-algebra's Grassmann manifold.