Classifies good involutions in conjugation subquandles and racks.
problem Classifying quandles with good involutions for applications in surface-knot theory.
method Study of subquandles of conjugation quandles, including core quandles; analysis of good involutions of faithful racks.
result Sharp bounds on the number of good involutions of racks in these families.
The study describes good involutions in quandles and Alexander quandles.
problem Characterizing and enumerating good involutions in quandles and Alexander quandles.
method Completely describing good involutions of free and subquandles of twisted conjugation quandles of groups, including Alexander quandles.
result Explicit mappings for good involutions of linear quandles up to order 23.
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.
We study the structure of finite quandles in terms of subquandles. Every finite quandle Q decomposes in a natural way as a union of disjoint Q-complemented subquandles; this decomposition coincides with the usual orbit decomposition of Q. Conversely, the structure of a finite quandle with a given orbit decomposit…
The paper studies quandles of hyperbolic 3-space isometries.
problem Investigating quandles of hyperbolic 3-space isometries.
method Introducing a new quandle Q(Γ,γ) and constructing a canonical map to the conjugate quandle. result The canonical map from Q(Γ,γ) to the conjugate quandle is injective and has a discrete image. We give a formula of the connected component decomposition of the Alexander quandle: Z[t±1]/(f1(t),…,fk(t))=⨆i=0a−1Orb(i), where a=gcd(f1(1),…,fk(1)). We show that the connected component Orb(i) is isomorphic to Z[t±1]/J with an expli…
We study the quandle counting invariant for a certain family of finite quandles with trivial orbit subquandles. We show how these invariants determine the linking number of classical two-component links up to sign.
The fundamental n-quandles of links are residually finite for n ≥ 2.
problem Residual finiteness of fundamental n-quandles of oriented links.
method Investigation of residual finiteness and subquandle separability of quandles; use of Winker's work on 3-sphere branched covers.
result Fundamental n-quandles of oriented links are residually finite for each n ≥ 2.
New polynomials defined for quandle structures, enhancing graph invariants.
problem Enhancing the counting invariant for spatial graphs and handlebody-links.
method Introducing quandle polynomials and G-family polynomials for quandles, defining enhancements for invariants.
result New enhancements of the G-family counting invariant for trivalent spatial graphs and handlebody-links.
We prove that the fundamental quandle of the trefoil knot is isomorphic to the projective primitive subquandle of transvections of the symplectic space Z⊕Z. The last quandle can be identified with the Dehn quandle of the torus and the cord quandle on a 2-sphere with four punctures. We also show that the fund…
Study of profinite quandles with constructions and characterizations.
problem Characterizing and constructing profinite quandles.
method Several constructions and characterizations of profinite quandles from profinite groups and other quandles.
result Characterization of algebraically connected profinite quandles in terms of $\widehat{\Inn(Q)}$.
This paper constructs quandles with abelian inner automorphism groups from graphs, proving their homogeneity.
problem Finding quandles with specific automorphism properties.
method Starting from simple graphs, the paper constructs quandles with abelian inner automorphism groups and proves their homogeneity.
result Homogeneous quandles with abelian inner automorphism groups are constructed from vertex-transitive graphs.
New manifolds found without interior conjugate points.
problem Existence of interior conjugate points in hyperbolic manifolds.
method Construction of non-trapping asymptotically hyperbolic manifolds.
result Found manifolds without 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. 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.
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…
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.
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. Proves robust transitivity for geodesic flows from metrics with conjugate points.
problem Transitivity of geodesic flows from metrics with conjugate points.
method General criterion for robust transitivity of partially hyperbolic geodesic flows.
result First example of a C2 open set of Riemannian metrics with conjugate points and transitive geodesic flow. Existence of a conjugate point proven on 3D ellipsoid.
problem Existence of conjugate points in incompressible Euler flow on 3D ellipsoid.
method Volume-preserving diffeomorphism group, Misiolek curvature criterion.
result Existence of a conjugate point on 3D ellipsoid.
The paper generalizes a mean value theorem for solutions of the ultrahyperbolic equation.
problem Establishing a mean value property for solutions of the ultrahyperbolic equation.
method Using conformal maps of the pseudo-Euclidean space of signature 2+2, the paper extends Asgeirsson's theorem to a more general class of pairs of curves.
result The mean value property is proven for non-degenerate conjugate conics, including conjugate circles, hyperbolae, parabolae, and line-empty pairs.
Generalizes Fenchel conjugation to nonlinear functions on arbitrary sets.
problem Extending Fenchel conjugation to functions on arbitrary sets without structure.
method Replacing linear test functions with nonlinear ones, investigating properties including biconjugation.
result Derived further results on smooth manifolds and Lie groups, relating to convexity.