Improves predictive performance of nested dichotomies.
problem Improving the performance of multi-class classification problems.
method A simple, general method for improving nested dichotomies produced by random subset selection techniques.
result Improves root mean squared error of nested dichotomies.
Study on curvature decay in steady Ricci solitons, proving dichotomy.
problem Curvature decay in steady Ricci solitons.
method Established a dichotomy for curvature decay in specific types of solitons.
result Proved a dichotomy on curvature decay for certain steady Ricci solitons.
Extends Hopf-Tsuji-Sullivan dichotomy to higher rank groups and applies to Anosov subgroups.
problem Understanding discrete subgroups of semisimple real algebraic groups.
method Establishes an extension of the Hopf-Tsuji-Sullivan dichotomy and applies it to Anosov subgroups.
result Anosov subgroups exhibit different phenomena depending on the rank of the group.
Discusses the tight versus overtwisted dichotomy in 3D contact geometry.
problem Understanding the tight versus overtwisted dichotomy in 3D contact geometry.
method Reviews Eliashberg's seminal work and contributions to the theory.
result Explains the genesis and importance of the tight versus overtwisted dichotomy.
Nested dichotomies for multiclass tasks often fail to calibrate probabilities.
problem Poor probability calibration in nested dichotomies for multiclass classification.
method Transforming multiclass problems into binary ones using a tree structure, and applying various calibration strategies.
result Improving accuracy and log-loss by calibrating both internal base models and the nested dichotomy structure.
The study proves a dichotomy for minimal hypersurfaces in certain thick manifolds.
problem Characterizing minimal hypersurfaces in thick manifolds.
method Generalizes Yau's conjecture using bounded geometry or thick at infinity conditions.
result Either infinitely many saddle points or no minimal hypersurfaces exist.
Study of Matsumoto maps on foliated bundles over hyperbolic manifolds.
problem Characterizing ergodic harmonic measures on foliated bundles.
method Analysis of actions of hyperbolic manifold groups on the circle.
result Suspension of actions with non-discrete images cannot admit Matsumoto maps of type I.
We offer the following explanation of the statement of the Kuratowski graph planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to every cell there corresponds a unique cell with the complementary set of vertices. Then …
Uniform boundedness of Riesz transforms on Riemannian manifolds is established with a dichotomy.
problem Establishing uniform boundedness of Riesz transforms on Riemannian manifolds.
method Constructing a complete Riemannian manifold M to demonstrate the dichotomy. result A dichotomy concerning uniform boundedness of Riesz transforms on Riemannian manifolds.
A system of nested dichotomies is a method of decomposing a multi-class problem into a collection of binary problems. Such a system recursively splits the set of classes into two subsets, and trains a binary classifier to distinguish between each subset. Even though ensembles of nested dichotomies with random structure…
TQFT invariants are either easy or hard to compute, depending on the TQFT type.
problem Computing TQFT invariants on closed 3-manifolds.
method Application of a dichotomy result for weighted constraint satisfaction problems over C.
result TQFT invariants are either solvable in polynomial time or #P-hard. The paper explores actions on metric spaces similar to 3D manifolds, proving rigidity results.
problem Rigidity of actions on metric spaces similar to 3D manifolds.
method Reexamined isometry groups of geometric 3-manifolds, considered homomorphisms to them, established a dichotomy.
result Established a dichotomy between finite image or infinite volume of quotient spaces.
The paper studies contact topology submanifolds and their rigidity.
problem Contact topology submanifolds rigidity and degeneracy.
method Contact coisotropic submanifolds, pseudo-metric, Chekanov type pseudo-metric.
result A contact topology analogue to Chekanov's dichotomy of degeneracy.
Anosov groups' measures on limit sets are uniquely determined by their dimension.
problem Characterizing measures on limit sets of Anosov groups.
method Higher rank Hopf-Tsuji-Sullivan dichotomy for maximal diagonal actions.
result Uniqueness of Γ-conformal measures for critical dimensions. The paper explores Anosov flows on torus bundles and provides new insights.
problem Investigating Anosov flows on affine torus bundles and their properties.
method Surgery type construction of Anosov flows and dichotomy result for fiberwise Anosov flows.
result New insights into Anosov flows, including non-transitive flows in odd dimensions.
New theory shows perishable goods markets are more stable and efficient.
problem Lower stability and efficiency of markets for re-tradable assets compared to perishable goods.
method Reformulation of no-trade and no-arbitrage theorems in neoclassical finance.
result Perishable goods markets exhibit higher stability and efficiency.
The paper studies ergodicity of flows on subspaces, generalizing earlier work.
problem Ergodicity of flows on subspaces of higher rank groups.
method Analyzes one-parameter diagonalizable subgroups of connected semisimple groups acting on homogeneous spaces.
result Obtains an ergodicity criterion similar to Hopf-Tsuji-Sullivan for general Anosov subgroups.
Classifies actions of groups on hyperbolic spaces, proving dichotomy.
problem Classifying actions of groups on hyperbolic spaces.
method Formalization using Borel equivalence relations, focusing on non-elementary actions without fixed points at infinity.
result For every countable group G, either all general type actions can be classified by an explicit invariant or they are unclassifiable in a strong sense. Study homotopy groups of open books and their pages, pages, and bindings.
problem Homotopy groups of open books and their components.
method Homotopy theoretic conditions on monodromy, integral and rational loop space decompositions.
result Integral and rational loop space decompositions for open books under specific conditions.
Study zippers in hyperbolic 3-manifolds, proving fixed point dichotomy.
problem Existence and behavior of elements in hyperbolic 3-manifold groups.
method Action analysis on minimal zippers, proving fixed point dichotomy.
result Every nontrivial element either fixes a unique point in each tree or acts freely on both.
The study quantifies how many objects can be linearly classified under all views.
problem Understanding the expressivity of group-equivariant representations.
method Generalization of Cover's Function Counting Theorem to quantify separable dichotomies.
result The fraction of separable dichotomies is determined by the fixed space dimension of the group action.
We study cohomological properties of complex manifolds. In particular, under suitable metric conditions, we extend to higher dimensions a result by A. Teleman, which provides an upper bound for the Bott-Chern cohomology in terms of Betti numbers for compact complex surfaces according to the dichotomy b1 even or odd.
New geometric structures defined in contact metric geometry.
problem Introducing new geometric structures in almost contact metric geometry.
method Defining locally conformal almost generalized f-cosymplectic manifolds and deriving integrability conditions. result Dimensional dichotomy: transverse components in dimension 3, proportional to η in higher dimensions.
Recent work of M. Yoshinaga shows that in some instances certain higher homotopy groups of arrangements map onto non-resonant homology. This is in contrast to the usual Hurewicz map to untwisted homology, which is always the zero homomorphism in degree greater than one. In this work we examine this dichotomy, generaliz…
Minimal submanifolds either fill space or are confined with geometric restrictions.
problem Understanding the behavior of minimal submanifolds in space.
method A dichotomy principle and volume doubling theorem.
result Quantitative restrictions on confined minimal submanifolds, including volume growth and optimal density rates.
Study dynamics of automorphisms on cubic surfaces and their connection to Painlevé 6.
problem Dynamics of holomorphic automorphisms on cubic surfaces and their relation to Painlevé 6.
method Defined Julia and Fatou sets, studied locally discrete and non-discrete dynamics, and proved existence of non-empty Fatou and Julia sets.
result Existence of non-empty Fatou and Julia sets for the group action.
The paper defines flexible domains for minimal surfaces in Euclidean spaces and explores their properties.
problem Understanding the flexibility of domains in Euclidean spaces for minimal surfaces.
method Investigates the concept of flexibility in terms of minimal surfaces contained in domains.
result Defines flexible domains and shows how they can be approximated by minimal immersions.
We study the boundary of an affine invariant submanifold of a stratum of translation surfaces in a partial compactification consisting of all finite area Abelian differentials over nodal Riemann surfaces, modulo zero area components. The main result is a formula for the tangent space to the boundary. We also prove fini…
Study free energy in spherical spin glasses, proving universality dichotomy.
problem Analyzing free energy in spherical spin glass models with different tail exponents.
method Introduced a tail-adapted normalization and used universality dichotomy.
result Sharp universality dichotomy for free energy across different tail exponents.
The paper develops a theory of conformal density at infinity for groups with contracting elements.
problem Understanding conformal dynamics at infinity for groups with contracting elements.
method Introducing a class of convergence boundary and establishing the basic theory of conformal density on it.
result Unified theory of conformal density on various boundaries for different types of groups.
The paper studies proper discontinuity of actions on Weyl chamber flow spaces.
problem Properly discontinuous actions on Weyl chamber flow spaces for transverse subgroups.
method Analyzes limit sets and quotient spaces, introduces growth indicators and conformal measures.
result Establishes ergodic dichotomy for Weyl chamber flow and introduces new measures.
New concept of regular separation for ODEs leads to improved Hardy field results.
problem Understanding solutions of definable ODEs with specific properties.
method Introducing regular separation and proving its implications for ODEs and vector fields.
result The regular separation property leads to improved Hardy field results and non-empty sets of trajectories.
Given a harmonic measure of a hyperbolic lamination on a compact metric space, a positive harmonic function is defined on the universal cover of a typical leaves. We discuss some properties of this function. Especially if all the leaves are hyperbolic, ergodic harmonic measures are divided into two classes.
This work refines Cover's theory for binary classification on low-dimensional data.
problem The challenge of analyzing how low-dimensional data structures affect classification models.
method Refines Cover's function-counting theory to account for low-dimensional data structure.
result Derives dichotomy counts and analyzes the impact of data structure on classification models.
We establish an integral test describing the exact cut-off between recurrence and transience for normally reflected Brownian motion in certain unbounded domains in a class of warped product manifolds. Besides extending a previous result by R. Pinsky, who treated the case in which the ambient space is flat, our result r…
We study the SL(2,R)-infimal lengths of simple closed curves on half-translation surfaces. Our main result is a characterization of Veech surfaces in terms of these lengths. We also revisit the "no small virtual triangles" theorem of Smillie and Weiss and establish the following dichotomy: the virtual triangle area spe…
Study traveling waves in hyperbolic space for Fisher-KPP equations.
problem Understanding wave behavior in hyperbolic space for Fisher-KPP equations.
method Analyzes the Cauchy problem in hyperbolic space for heat equation with Fisher-KPP forcing term.
result Proves new results on the dichotomy of solution propagation or vanishing based on diffusion and reaction strength.
Breiman's data analysis dichotomy is outdated, offering a third approach: mechanistic models.
problem Data analysis dichotomy between data modelers and algorithmic modelers.
method Interpolating between simple interpretable models and flexible function approximations using mechanistic models.
result Flexible, interpretable, and scientifically-informed hybrids can provide accurate and robust predictions.
The paper studies topological properties of Ricci shrinkers using weighted L2 cohomology.
problem Proving topological results for smooth gradient Ricci shrinkers.
method Weighted L2 cohomology and extensions to mean curvature flow self-shrinkers. result Establishes upper bounds for Betti numbers, vanishing theorem for cohomology, and dichotomy for ends.
In this paper a conformal classification of three dimensional left-invariant sub-Riemannian contact structures is carried out; in particular we will prove the following dichotomy: either a structure is locally conformal to the Heisenberg group H3, or its conformal classification coincides with the metric one…
Average teaching complexity for locating target regions among halfspace intersections is Θ(d).
problem Teaching the location of a target region among intersections of halfspaces.
method Novel insights from computational geometry to count convex polytopes and faces.
result Average-case teaching complexity is Θ(d), contrasting with Θ(n) worst-case complexity.
Optimal control problem for firm cash flow with dividend and capital injection strategies.
problem Maximizing dividends while managing capital injections in a firm's cash flow.
method Proved two optimal strategies: mean-reverting dividends with capital injections or no injections until ruin.
result Optimal strategies are dichotomous: either mean-reverting dividends with injections or no injections.
In this paper we extend the local scalar curvature rigidity result in [6] to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper [10]. We obtain the local scalar curvature rigidi…
The Higgs field growth is studied on special geometric spaces, confirming a conjecture.
problem Growth of the Higgs field in special geometric spaces.
method Analyzing θ-Kapustin-Witten equations on ALX spaces. result Finite energy solutions on ALE and ALF instantons have vanishing commutator and flat connection.
Two extremal classes of acyclic groups are discussed. For an arbitrary group G, there is always a homomorphism from an acyclic group of cohomological dimension 2 onto the maximum perfect subgroup of G, and there is always an embedding of G in a binate (hence acyclic) group. In the other direction, there are no nontrivi…
Given a nonconvex function that is an average of n smooth functions, we design stochastic first-order methods to find its approximate stationary points. The convergence of our new methods depends on the smallest (negative) eigenvalue −σ of the Hessian, a parameter that describes how nonconvex the function is. Our m…
The Kapustin-Witten equations on R^4 are equations for a pair of connection on the product principle SU(2) bundle and 1-form with values in the product Lie algebra bundle. The 1-form is the Higgs field. A dichotomy is proved to the effect that either the averaged norm of the Higgs field on large radius spheres grows fa…
The Tait conjecture states that reduced alternating diagrams of links in S^3 have the minimal number of crossings. It has been proved in 1987 by M. Thistlethwaite, L.H. Kauffman and K. Murasugi studying the Jones polynomial. In this paper we prove an analogous result for alternating links in S^1xS^2 giving a complete a…