Unified approach to measure expanders using finite graphs.
problem Constructing and understanding measure expanders.
method Defining and analyzing finite graphs approximating actions on measure spaces.
result Graphs form expanders if and only if the action is expanding in measure.
New expanders found using origami surfaces with spectral gap.
problem Constructing expanders with spectral gap on surfaces of arbitrary genus.
method Affine actions on origami surfaces to achieve spectral gap.
result New expanders distinct from classical ones.
We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with low…
Using optimal transport we study some dynamical properties of expanding circle maps acting on measures by push-forward. Using the definition of the tangent space to the space of measures introduced by Gigli, their derivative at the unique absolutely continuous invariant measure is computed. In particular it is shown th…
This paper classifies all expanding Ricci solitons on surfaces.
problem Identifying all expanding Ricci solitons on surfaces.
method Developed a Ricci flow existence theory and used uniqueness theory to classify solitons.
result Classified all expanding Ricci solitons on surfaces.
Study examines risk premium convergence rates in risk sharing contracts.
problem Analyzing risk premium convergence rates in risk sharing contracts.
method Examines the limiting behavior of risk premium associated with Pareto optimal risk sharing contracts under general law-invariant risk measures.
result Risk premium convergence rate is typically n1/2, not n. Study shows twist tori equidistribute in moduli space, with other families having singular distributions.
problem Statistical behavior of twist tori in moduli space of hyperbolic surfaces.
method Analyzing expanding families of twist tori and their limiting distributions.
result Equidistribution of twist tori to a Lebesgue measure, with other families having singular distributions.
The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.
problem Solving the (p,q)-Christoffel-Minkowski problem.
method Investigating the problem via an expanding curvature flow.
result Existence and uniqueness of smooth solutions to the (p,q)-Christoffel-Minkowski problem.
Study stationary measures and orbit closures for non-abelian actions on surfaces.
problem Classify stationary measures and orbit closures for non-abelian action on a surface.
method Use a finite verifiable average growth condition and results from Brown and Rodriguez Hertz.
result Show that under certain conditions, the only nonatomic stationary measure is the given smooth invariant measure, and every orbit closure is either finite or dense.
The study proves topological rigidity for certain geometric shapes using Poincaré inequalities.
problem Proving topological rigidity for translators and self-expanders in mean curvature flow.
method Abstract structure theorem for weighted manifolds, Poincaré inequality, and topological control.
result Full topological control on translators and self-expanders under stability or curvature assumptions.
The paper proves conditions for non-uniform expansion in partially hyperbolic systems.
problem Conditions for non-uniform expansion in partially hyperbolic systems.
method Analysis of Lyapunov exponents and dominated splittings.
result Existence of physical SRB measure under specific conditions.
Future stability of FLRW solutions in expanding 3D space is shown for compact perturbations.
problem Future stability of expanding FLRW solutions with spatial topology R^3.
method Nonlinear stability analysis of spherically symmetric perturbations.
result Decay rates of energy momentum tensor components compared to Minkowski space.
Expands robust profit opportunities to include distributional uncertainty.
problem Distributional uncertainty in financial markets.
method Formulates infinite dimensional primal problems, simplifies to finite dimensional dual problems using Wasserstein distance.
result Distributional uncertainty can enhance robustness of profit opportunities.
Expands learning paradigm to stochastic orders using Choquet-Toland distance and Variational Dominance Criterion.
problem Learning high-dimensional distributions with stochastic orders.
method Introduces Choquet-Toland distance and Variational Dominance Criterion, uses input convex maxout networks (ICMNs).
result Proposes surrogates for Choquet-Toland distance and Variational Dominance Criterion with parametric rates.
Expands newsvendor model with moment constraints using Wasserstein distance.
problem Optimizing order quantity under distributional ambiguity.
method Formulates infinite dimensional primal problem, derives finite dimensional dual problem using problem of moments duality.
result Distributional ambiguity affects optimal order quantity and profits/costs.
Convex optimization with expander matrices improves sparse recovery efficiency.
problem Sparse recovery from linear measurements using expander matrices.
method Use of expander matrices for linear sketches in convex optimization to recover block-sparse matrices.
result The recovery error can be expressed in terms of the model-based norm, ensuring the solution is within the model.
Paper develops a decoder for sparse codes without encoder matrix, achieving optimal recovery.
problem Designing a decoder for sparse codes from linear measurements alone.
method Matrix factorization to recover encoder and sparse coding matrices from measurements.
result Decoder-Expander Based Factorisation recovers encoder and sparse coding matrix at optimal measurement rate with high probability.
Constructs flow lines connecting unstable to stable self-expanders.
problem Existence of monotone Morse flow lines for expander functionals.
method Constructs a singular Morse flow line connecting unstable to stable self-expanders.
result Constructs a monotone flow line with a small singular set.
The paper extends rigidity results for λ-self-expanders to hyperplanes, spheres, and cylinders.
problem Characterizing λ-self-expanders as hyperplanes, spheres, and cylinders. method Extending results on self-expanders to λ-self-expanders, proving rigidity results. result Characterizes hyperplanes, spheres, and cylinders as λ-self-expanders. The study explores dilating set properties across Euclidean and hyperbolic geometries.
problem Distributional properties of dilating sets under projection.
method Covering maps and unit tangent bundles, focusing on manifolds of constant curvature.
result Established a precise asymptotic expansion for averages along expanding translates of homogeneous curves in hyperbolic surfaces.
Solves Ricci flow on Riemann surfaces with measure initial data.
problem Existence and smoothness of Ricci flow on Riemann surfaces.
method Formulation and solution of existence problem using Ricci flow.
result New examples of nongradient expanding Ricci solitons.
New self-expander found between two given asymptotic ones.
problem Finding new self-expanders between given asymptotic ones.
method Developed a min-max theory for asymptotically conical self-expanders of mean curvature flow.
result Existence of a new asymptotically conical self-expander trapped between two given ones.
Unique expanders found with vanishing entropy.
problem Finding unique expanders with specific entropy properties.
method Adapting White's work and using Bernstein-Wang results.
result Generic uniqueness of expanders with vanishing relative entropy.
Rotational symmetry proven for certain self-expanders.
problem Analyzing self-expanders with decaying principal curvatures.
method Proved a Liouville-type theorem and applied it.
result Rotational symmetry for specific self-expanders.
No expanding breathers found in noncompact Ricci flows with certain curvature conditions.
problem Finding expanding breathers in noncompact Ricci flows.
method Curvature positivity conditions (weak PIC-2 or nonnegative bisectional curvature).
result Complete noncompact expanding breathers are gradient solitons.
Transformers can interpolate between arbitrary measures.
problem Understanding the expressive power of Transformers as measure-to-measure maps.
method Provided an explicit choice of parameters for a single Transformer to match N arbitrary input measures to N arbitrary target measures.
result A single Transformer can interpolate between arbitrary measures.
Sharp curvature estimates for expanding Ricci solitons in various dimensions.
problem Estimating curvature bounds for expanding Ricci solitons.
method Sharp lower and upper bounds derived for scalar curvature under specific conditions.
result Sharp curvature estimates provided for expanding Ricci solitons in dimensions three and four.
Study shows smooth manifold of self-expanders in mean curvature flow.
problem Understanding the structure of self-expanders in mean curvature flow.
method Analyzes asymptotically conical self-expanders as a smooth Banach manifold.
result Non-degenerate self-expanders are generic.
Study on measurable pseudo-Anosov maps on surfaces.
problem Characterize dynamics of pseudo-Anosov maps on surfaces.
method Analyze measurable pseudo-Anosov homeomorphisms with specific properties.
result Prove transitivity, dense periodic points, sensitivity, and ergodicity.
Study expanding solitons on complex Lie groups with specific algebraic structures.
problem Investigate expanding solitons on complex Lie groups with left-invariant metrics.
method Analyze the algebraic structure of complex Lie groups and their decompositions.
result Show that the Lie algebras of complex Lie groups decompose into semidirect products.
Compactness proven for specific types of self-expanders in mean curvature flow.
problem Proving compactness for asymptotically conical self-expanders of mean curvature flow.
method Analyzing families of self-expanders and showing compactness in locally smooth topology.
result Properness of the projection map for specified classes of self-expanders.
New expanders for mean curvature flow contradict genus-reduction conjecture.
problem Contradicting Ilmanen's genus-reduction conjecture for mean curvature flow.
method Construct new expanders asymptotic to cones arising from shrinkers.
result Existence of expanders of arbitrarily large genus.
New degree theory proves existence of solitons on 4D manifolds.
problem Existence of gradient expanding solitons on 4D manifolds.
method Developed new degree theory for 4D, asymptotically conical gradient expanding solitons.
result Existence of solitons asymptotic to any cone over S^3 with non-negative scalar curvature.
Study semiclassical measures on complex hyperbolic quotients, identifying measure supports.
problem Understanding Laplacian eigenfunctions on complex hyperbolic quotients.
method Combining fractal uncertainty principle and Ratner theory to analyze measure supports.
result Semiclassical measures support is either cosphere bundle or a compact submanifold.
New method detects changes online with bounds on delay.
problem Detecting changes in data streams efficiently.
method Maximizes discrepancy between pre-change and post-change distributions.
result Non-asymptotic bounds on average running length and detection delay.
New expanding Ricci solitons found starting in dimension four.
problem Finding expanding Ricci solitons in specific dimensions.
method Constructing gradient expanding Ricci solitons asymptotic to cones and on trivial vector bundles.
result Continuous families of expanding Ricci solitons on products of Einstein manifolds.
Stable expanding solitons with positive curvature decay proven.
problem Stability of expanding gradient Ricci solitons with positive curvature.
method Proving weak stability with quadratic curvature decay.
result Proven stability of expanding solitons with positive curvature.
Study on the spectrum of drift Laplacian on Ricci expanders.
problem Analyzing the spectrum of the drift Laplacian on Ricci expanders.
method Investigation of discrete spectrum under proper potential function, asymptotic behavior of potential function, and computation of eigenvalues.
result Discrete spectrum of the drift Laplacian on Ricci expanders with bounded Ricci curvature.
Expander graphs have been a focus of attention in computer science in the last four decades. In recent years a high dimensional theory of expanders is emerging. There are several possible generalizations of the theory of expansion to simplicial complexes, among them stand out coboundary expansion and topological expand…
The paper classifies expanding gradient Yamabe solitons based on scalar curvature.
problem Classifying expanding gradient Yamabe solitons based on scalar curvature.
method Rigorous analysis of scalar curvature in both cases: greater than and less than the soliton constant.
result Complete classification of nontrivial complete expanding gradient Yamabe solitons.
Constructs self-expanders of positive genus for cones in R^3.
problem Creating self-expanders of positive genus for cones in R^3.
method Constructs self-expanders asymptotic to cones, uses mean curvature flow.
result Constructs self-expanders with unbounded genus asymptotic to a rotationally symmetric cone.
Researchers expand on best subset selection theory, identifying key complexities.
problem Understanding model selection performance in high-dimensional sparse linear regression.
method Analyzing residualized signals, orthogonality, and spurious projections to establish margin conditions.
result Established necessary and sufficient margin conditions for BSS model consistency.
Study of complete space-like self-expanders in Minkovski space.
problem Characterize complete space-like self-expanders in Minkovski space.
method Use of maximum principle of Omori-Yau type to prove rigidity theorems.
result Classification of 2-dimensional complete space-like self-expanders with constant squared norm of the second fundamental form.
Study finds unique self-expanders for mean curvature flow.
problem Finding solutions to mean curvature flow.
method Derived equation based on generalized Lawson-Osserman cone and modified equilibria theory.
result Existence and uniqueness of self-expanders proved.
New theory explores high-dimensional expanders.
problem Understanding high-dimensional expanders.
method Exploring new mathematical and computational approaches.
result Developed new methods to study high-dimensional expanders.
Study cohomogeneity one expanding Ricci solitons on specific topologies.
problem Characterize and analyze cohomogeneity one expanding Ricci solitons.
method Analyze ODEs, define expander degree, calculate cohomogeneity one expander degree.
result Reconstruct and calculate cohomogeneity one expander degree for specific topologies.
Study improves confidence measures in medical imaging pipelines by addressing bias.
problem Bias in metric-based imaging pipelines compromises the efficiency of prediction intervals.
method Formalized symmetric and asymmetric CP formulations, analyzed bias effects, and validated empirically.
result Symmetric intervals are inflated by bias, while asymmetric intervals remain unaffected.
The paper examines properties and rigidity of self-expanders in Euclidean space.
problem Characterizing and estimating properties of self-expanders in Euclidean space.
method Analyzing mean curvature flow, volume growths, and stability of self-expanders.
result Proves the uniqueness of certain self-expanders in 3D space.