Theory of cone fields using relaxed orbits for Lyapounov functions and temporal functions.
problem Existence of Lyapounov functions and temporal functions in cone fields.
method Using a notion of relaxed orbits based on cone enlargements, in the spirit of space-time geometry.
result Generalization of equivalence between stable causality and the existence of temporal functions.
We give a sharp lower bound for the number of geometrically distinct contractible periodic orbits of dynamically convex Reeb flows on prequantizations of symplectic manifolds that are not aspherical. Several consequences of this result are obtained, like a new proof that every bumpy Finsler metric on Sn carries at l…
Random polynomial dynamical systems often have negative Lyapunov exponents.
problem Understanding the behavior of random polynomial dynamical systems.
method Investigation of i.i.d. random complex dynamical systems generated by probability measures.
result For a generic system, the Lyapunov exponent of almost every sequence of maps is negative for most initial values.
Using the theory of group action, we first introduce the concept of the automorphism group of an exponential family or a graphical model, thus formalizing the general notion of symmetry of a probabilistic model. This automorphism group provides a precise mathematical framework for lifted inference in the general expone…
Local bifurcation theory typically deals with the response of a degenerate but isolated equilibrium state or periodic orbit of a dynamical system to perturbations controlled by one or more independent parameters, and characteristically uses tools from singularity theory. There are many situations, however, in which the…
Study limits of adjoint orbits for Lie groups, describing nilpotent orbits.
problem Understanding limits of adjoint orbits for Lie groups.
method Systematic and topological study of limits of continuous families of adjoint orbits for non-compact simple Lie groups.
result Explicit description of nilpotent orbits in terms of Richardson orbits for hyperbolic semisimple elements.
Geodesic orbit Riemannian structures on R^n characterized.
problem Characterizing Riemannian manifolds with geodesic orbits.
method Geometric and algebraic characterization of geodesic orbit manifolds diffeomorphic to R^n.
result Established structural properties of geodesic orbit manifolds.
We define the notion of the orbit group of a quandle via its connectivity and compute the orbit groups for some basic quandles. We also show that the orbit group counts the number of orbits of certain quandles.
The paper finds linked periodic orbits in disc homeomorphisms using braids.
problem Finding linked periodic orbits in disc homeomorphisms.
method Interpreting linking of orbits by induced braids and using forcing relations.
result New examples of linked orbits of periods at most 4 for pseudo-Anosov braid types.
The paper finds symplectic compactifications of coadjoint orbits.
problem Understanding symplectic structures on coadjoint orbits.
method Defined real analytic symplectomorphisms on subsets of coadjoint orbits.
result Coadjoint orbits of compact Lie algebras are symplectic compactifications of domains of cotangent bundles.
New insights into pseudo-Anosov flows with special periodic orbits.
problem Understanding pseudo-Anosov flows with periodic orbits in 3-manifolds.
method Analyzing the topological features corresponding to trees of scalloped regions and classifying flows with the same free homotopy data.
result Explicit examples of flows with the same free homotopy data but not orbit equivalent.
The study counts periodic orbits on smooth manifolds, adding ghost orbits for completeness.
problem Counting periodic orbits of vector fields on smooth closed manifolds.
method Enlarging the space of orbits to include ghost orbits, defining weight functions, and showing constancy under deformation.
result The weight function remains constant as the vector field moves and Γ deforms. Study properties of orbits of Hermann actions without commutability assumptions.
problem Investigate geometric properties of orbits of Hermann actions.
method Compute the second fundamental form and provide conditions for weak reflection and aridity.
result Sufficient conditions for weak reflection and aridity of orbits of Hermann action.
The paper characterizes contact 3-manifolds with closed Reeb orbits.
problem Characterizing contact 3-manifolds with closed Reeb orbits.
method Analyzing the action spectrum and minimal periods of Reeb orbits.
result A contact form with an action spectrum of rank 1 is uniquely determined by the minimal periods of its closed Reeb orbits.
The study characterizes geodesic orbit Riemannian spaces and their properties.
problem Characterizing geodesic orbit Riemannian spaces and their properties.
method Analyzing the structure of nilradical and radical of Lie algebra, discussing compact Lie group representations.
result Described the structure of nilradical and radical of Lie algebra of isometry group.
New conic relaxations improve sparse signal recovery with fewer observations.
problem Recovering sparse signals from noisy data.
method Comparing two semidefinite relaxations for sparse linear regression.
result Dong's relaxation requires fewer observations for exact recovery.
Study classifies compact geodesic orbit spaces with two isotropy components.
problem Characterizing geodesic orbit Riemannian spaces.
method Classification of spaces with specific isotropy properties.
result Classification of compact geodesic orbit spaces with two isotropy summands.
Smooth approximations for continuous functions on orbit spaces.
problem Approximating continuous functions on orbit spaces.
method Study of subcartesian spaces and proper Lie group actions.
result Continuous functions can be approximated by smooth functions.
Minimal orbits of semi-simple Lie groups are studied and related to invariant subspaces.
problem Characterizing minimal orbits of semi-simple Lie groups.
method Analyzing projective orbits induced by representations of semi-simple Lie groups and relating them to invariant subspaces of the underlying modules.
result Minimal orbits of semi-simple Lie groups are in bijection with minimal orbits of compact subgroups on invariant subspaces.
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.
Convex relaxations improve CNNs with fixed weights.
problem Improving CNNs with fixed weights.
method Convex relaxations for CNNs with fixed weights using second order cone programs.
result The relaxation recovers the global minimum under a planted model assumption.
New method improves neural network verification by considering multivariate input space of ReLU neurons.
problem Improving the effectiveness of neural network verification algorithms.
method A new tightened convex relaxation for ReLU neurons considering multivariate input space.
result Our convex relaxation is significantly stronger than the commonly used univariate-input relaxation.
New semidefinite relaxation improves robustness certification of neural networks.
problem Certifying robustness of neural networks against adversarial examples.
method Proposed a new semidefinite relaxation for certifying robustness of arbitrary ReLU networks.
result Our proposed relaxation is tighter than previous relaxations and produces meaningful robustness guarantees.
New Frobenius manifold structures found on Dicyclic group orbits.
problem Finding Frobenius manifold structures on orbits spaces of Dicyclic groups.
method Applying Dubrovin's method to Dicyclic groups.
result Dicyclic group orbits spaces acquire two Frobenius manifold structures.
Given a compact Riemannian manifold together with a group of isometries, we discuss MCF of the orbits and some applications: eg, finding minimal orbits. We then specialize to Lagrangian orbits in Kaehler manifolds. In particular, in the Kaehler-Einstein case we find a relation between MCF and moment maps which, for exa…
Study Kleinian groups using orbital functions and heat kernels.
problem Understanding Kleinian groups through orbital functions and heat kernels.
method Using the heat kernel approach developed in \cite{artmoiheatcounting1}.
result Developed a method to study Kleinian groups using orbital functions and heat kernels.
Study orbits in right triangles, deducing periodic billiard paths and classifying orbit closures.
problem Understanding periodic billiard paths in right triangles and orbit closures in strata of Abelian and quadratic differentials.
method Classifying orbit closures of rank at least two in hyperelliptic components of strata of Abelian and quadratic differentials.
result Computed orbit closures and deduced asymptotic number of periodic billiard trajectories in right triangles.
New criteria found for Willmore submanifolds in Lie group orbits.
problem Criteria for Willmore submanifolds in Lie group orbits.
method Criteria for Willmore submanifolds based on orbit type stratification.
result Found Willmore orbits in each stratified subset of orbit type.
Circle actions yield different orbit spaces.
problem Understanding differential structures on orbit spaces.
method Analyzing non-isomorphic linear circle actions.
result Non-diffeomorphic orbit spaces from different actions.
New regularizers tighten convex relaxation bounds for neural networks.
problem Large gap between certifiable and empirical robustness in neural networks.
method Two regularizers to train neural networks yielding tighter convex relaxation bounds.
result Higher certified accuracy with proposed regularizers.
Study orbit spaces of equivariant ANEs for proper actions of metrizable groups.
problem Understanding the extension properties of orbit spaces for proper actions.
method Analyzing equivariant absolute neighborhood extensors for proper G-spaces. result Proving conditions under which orbit spaces of metrizable G-orbits are ANEs. Study infinitesimally tight Lagrangian orbits in symplectic manifolds.
problem Understanding Lagrangian orbits in symplectic manifolds.
method Analyzing isotropic orbits of Lie group actions with equivariant moment maps.
result Examples of Lagrangian orbits in complex flag manifolds and cotangent bundles.
We produce infinitely many examples of Anosov flows in closed 3-manifolds where the set of periodic orbits is partitioned into two infinite subsets. In one subset every closed orbit is freely homotopic to infinitely other closed orbits of the flow. In the other subset every closed orbit is freely homotopic to only one …
Study of adjoint orbits in simplest non-trivial Lie algebra case.
problem Geometric properties of adjoint orbits in sl(2,R). method Analysis of adjoint orbits, showing three possibilities: hyperboloids or cones.
result Just three possibilities for adjoint orbits: hyperboloids or cones.
The classification of G-spaces by Palais is refined for the case where the orbit space satisfies certain mild topological hypotheses. It is shown that when a sequence of such orbit spaces is "close" to a limit orbit space, in some suitable sense, within a larger ambient orbit space, the G-spaces in the tail of the sequ…
We list up all the possible local orbit types of hyperbolic or elliptic orbits for the isotropy representations of semisimple pseudo-Riemannian symmetric spaces. It is key to give a recipe to determine the local orbit types of hyperbolic principal orbits by using three kind of restricted root systems and Satake diagram…
Improved neural network robustness certification through tighter convex relaxations.
problem Certifying neural network robustness to perturbed and adversarial inputs.
method Exploiting ReLU network structure, novel partition-based certification procedure.
result Tightens existing linear programming relaxations to achieve zero relaxation error asymptotically.
Paper presents a novel orbit determination method for spacecraft clusters.
problem Orbit determination for spacecraft clusters with noisy and non-linear observations.
method Kernel embedding techniques for learning orbits from range-rate observations.
result The method can accurately estimate orbits and identify individual satellites.
Geodesic graphs for special Finsler metrics on spheres are studied.
problem Characterizing geodesic orbit Finsler metrics on spheres.
method Explicit constructions and group extensions.
result Not all projective spaces admit invariant Finsler metrics.
Classifies actions on Minkowski space up to orbit equivalence.
problem Classifying actions on Minkowski space up to orbit equivalence.
method Classifying actions up to orbit equivalence, providing representations and orbit spaces.
result Orbits and orbit spaces determined for proper actions.
Classifies finite orbits of mapping class group action on character varieties.
problem Classifying finite orbits of mapping class group action on character varieties of punctured spheres.
method Inductive proof using Lisovyy--Tykhyy's classification for 4-punctured spheres as base case.
result Proves no finite orbits for 7-punctured spheres and unique 1-parameter family for 6-punctured spheres.
A faster X-ray CT image reconstruction method using relaxed linearized algorithms.
problem Reduced X-ray dose while maintaining image quality in CT scans.
method Relaxed linearized augmented Lagrangian (AL) method with over-relaxation.
result The proposed method is about twice as fast as existing unrelaxed fast algorithms.
We study holomorphic extensions of Matsuki orbits in complex Grassmannians.
problem Analyzing the analytic continuation of Matsuki orbits in complex Grassmannians.
method Using Rossi's theory of holomorphic extension and the holomorphic fiber bundle structure, we establish that the envelope of holomorphy of each Matsuki orbit coincides biholomorphically with the containing K-orbit. result The envelope of holomorphy of each Matsuki orbit coincides biholomorphically with the containing K-orbit. The paper classifies geodesic orbit spaces with simple isotropy groups.
problem Classifying geodesic orbit spaces with simple isotropy groups.
method Classifying G-naturally reductive and G-geodesic orbit metrics on M. result Classification of geodesic orbit spaces with simple isotropy groups.
Abstract not provided enough details, focusing on vector bundles and orbits.
problem Understanding continuous representations of semisimple Lie groups.
method Not specified in the abstract.
result Not specified in the abstract.
Computer program finds flower-like periodic orbits in Kepler-Heisenberg problem.
problem Motion of a planet around a sun in the sub-Riemannian Heisenberg group.
method Monte Carlo optimization with a shooting method and a symplectic integrator.
result Discovery of a family of flower-like periodic orbits with new symmetry types.
Criterion for periodic orbits convergence proved.
problem Periodic orbits convergence criterion.
method Criterion for Benjamini-Schramm convergence of periodic orbits of Lie groups.
result Criterion for periodic orbits convergence proved.
We characterize isometric actions on compact Kaehler manifolds admitting a Lagrangian orbit, describing under which condition the Lagrangian orbit is unique. We furthermore give the complete classification of simple groups acting on the complex projective space with a Lagrangian orbit, and we give the explicit list of …