New findings on asymptotic property C in infinite dimensional spaces.
problem Understanding infinite dimensional spaces with infinite asymptotic dimension.
method Showed preservation of asymptotic property C in infinite products and introduced hyperbolic property C.
result Infinite products and restricted direct products of countable groups with finite asymptotic dimension have asymptotic property C.
Groups with certain properties act on metric spaces with asymptotic property C.
problem Understanding groups acting on metric spaces with specific properties.
method Analyzing quasi-stabilizers and asymptotic dimension.
result Groups with quasi-stabilizers of bounded asymptotic dimension have asymptotic property C.
The paper explores properties preserved by coarsely n-to-1 functions in metric spaces.
problem Investigating properties preserved by coarsely n-to-1 functions in metric spaces.
method Analyzing properties related to asymptotic dimension and its generalizations.
result The class of spaces with straight finite decomposition complexity coincides with the class of spaces of countable asymptotic dimension.
The paper proves stability of asymptotic property C in group extensions and products.
problem Stability of asymptotic property C in group extensions and products.
method Analyzes direct products, free products, and group extensions to prove asymptotic property C.
result Groups with asymptotic property C preserve this property under direct products and free products, and under certain group extensions.
New complexity notion connects finite decomposition and asymptotic property C.
problem Understanding and connecting different properties in metric spaces.
method Introducing finite APC-decomposition complexity and proving its implications.
result Finite APC-decomposition complexity implies property A for metric spaces.
New geometric property for metric spaces with infinite dimensions.
problem Classifying metric spaces with infinite asymptotic dimension.
method Introducing a new geometric property called 'complementary-finite asymptotic dimension' (coas-dim). Proving corresponding coarse invariant theorems.
result Established the geometric property and proved theorems related to it.
The paper proves extension theorems for various geometric properties of groups.
problem Various geometric properties of groups and their extensions.
method Proving extension theorems for asymptotic property C, finite decomposition complexity, and strict finite decomposition complexity.
result Groups with certain properties inherit weaker geometric properties.
We present the notion of asymptotically large depth for a metric space which is (a priory) weaker than having subexponential asymptotic dimension growth and (a priory) stronger than property A.
Study on finiteness properties of handlebody mapping class groups.
problem Understanding finiteness properties of asymptotically rigid handlebody groups.
method Introduced asymptotically rigid mapping class groups and determined their finiteness properties based on the space of ends of handlebodies.
result Homology of these groups coincides with stable homology of handlebody groups in some cases.
The paper analyzes network models with binary values and sub-Gamma noise, deriving asymptotic properties.
problem Analyzing network models with binary values and sub-Gamma noise.
method Derives asymptotic properties of network models with binary values and sub-Gamma noise.
result Established asymptotic consistency and normality of parameter estimators in network models.
This is a survey of the recent work in algorithmic and asymptotic properties of groups. I discuss Dehn functions of groups, complexity of the word problem, Higman embeddings, and constructions of finitely presented groups with extreme properties (monsters).
Proves existence of initial data with specific curvature properties for Einstein equations.
problem Finding initial data for Einstein equations with specific curvature properties.
method Proves existence of a class of initial data with finite ends, mild curvature conditions.
result Existence of trumpet solutions with non-constant mean curvature.
We study asymptotic properties of maximum likelihood estimators for Heston models based on continuous time observations of the log-price process. We distinguish three cases: subcritical (also called ergodic), critical and supercritical. In the subcritical case, asymptotic normality is proved for all the parameters, whi…
Study on special Kähler metrics with specific asymptotic properties.
problem Finding complete constant scalar curvature Kähler metrics with Poincaré-Mok-Yau asymptotic property.
method Use of Calabi's ansatz and moment construction.
result Provide examples of such metrics on non-compact Kähler manifolds.
Study on MLE growth rate for stable CIR process, proving consistency and normality.
problem Estimating the growth rate of a stable CIR process from continuous observations.
method Maximum likelihood estimation for a specific type of process.
result Strong consistency and asymptotic normality in subcritical and supercritical cases, asymptotic mixed normality in supercritical, open in critical case.
In this short note, we study the asymptotic property of Huisken's functional for mean curvature flow on the minimal submanifolds of Euclidean space. We prove that the limit of Huisken's functional equals to the extrinsic asymptotic volume ratio on the minimal submanifold of Euclidean space.
Study properties of Weingarten surfaces using curvature nets.
problem Characterizing Weingarten surfaces in 3D space.
method Analysis of curvature nets on central surfaces.
result Properties of Weingarten surfaces identified.
The paper studies properties of group relations induced by compatible coarse structures.
problem Properties of asymptotic resemblance relations on groups.
method Generalization of asymptotic dimension and introduction of set theoretic coupling.
result Groups with compatible coarse structures that admit a set theoretic coupling are asymptotic equivalent.
New topological property of self-shrinkers with small entropy.
problem Understanding the topological structure of self-shrinkers with small entropy.
method Analyzing asymptotically conical self-shrinkers with entropy ≤ cylinder entropy.
result The link of the asymptotic cone separates the unit sphere into two diffeomorphic components.
In this paper, we study the asymptotic behavior of the volume of spheres in metric measure spaces. We first introduce a general setting adapted to the study of asymptotic isoperimetry in a general class of metric measure spaces. We then introduce a notion of "being asymptotically isoperimetric" for a family of finite a…
We study some asymptotic properties of the sequences of symplectic Lefschetz pencils constructed by Donaldson. In particular we prove that the vanishing spheres of these pencils are, for large degree, conjugated under the action of the symplectomorphism group of the fiber. This implies the non-existence of homologicall…
Research preserves coarse property C and related dimensions through direct products.
problem Preserving properties in coarse geometry through direct products.
method Demonstrates preservation of coarse property C and related dimensions through finite coarse direct products.
result Coarse property C and related dimensions are preserved by direct products.
Defines CAMC discrete nets and their properties.
problem Understanding CAMC discrete nets and their properties.
method Defining CAMC discrete nets and proving properties.
result Properties of CAMC discrete nets are equivalent to properties of compatible interpolating quadrics.
The study finds monotonic properties of harmonic functions on 3-manifolds with a flat end.
problem Understanding harmonic functions on 3-manifolds with specific ends.
method Derives monotonic properties of positive harmonic functions on 3-manifolds with nonnegative scalar curvature and asymptotically flat ends.
result Rigidity characterization of spatial Schwarzschild manifolds with two ends.
Sharp thresholds found for graph properties in random graphs.
problem Understanding structural properties of random graphs.
method Analysis of Erdös--Rényi random graph model G(n,p).
result Sharp thresholds for AS and CFS properties in random graphs.
Geodesic completeness and optimal Sobolev index proven for Minkowski spacetimes.
problem Geodesic completeness and optimal Sobolev index for Minkowski spacetimes.
method Null non-trapping condition and real principal type estimate.
result Optimal Sobolev index proven for asymptotically Minkowski spacetimes.
Formula derived for a magnetic line invariant.
problem Solving MHD problems with two-scale mean fields.
method Combinatorial definition of M3 invariant for three-component links. result Proven formula for M invariant verified for simple cases. When working with asymptotically hyperbolic initial data sets for general relativity it is convenient to assume certain simplifying properties. We prove that the subset of initial data sets with such properties is dense in the set of physically reasonable asymptotically hyperbolic initial data sets. More specifically, …
Introduces regular finite decomposition complexity for metric families.
problem Metric families with finite asymptotic dimension.
method Introduces regular finite decomposition complexity (RFDC) and shows it implies finite decomposition complexity (FDC).
result RFDC has permanence properties including Finite Quotient Permanence.
Defines coarse versions of property C and decomposition complexity.
problem Tackles coarse versions of property C and decomposition complexity.
method Uses coarse spaces to define coarse versions of asymptotic property C and decomposition complexity.
result Proves coarse property C implies coarse property A and shares features with metric analogs.
Let G be a finitely generated group with a given word metric. The asymptotic density of elements in G that have a particular property P is defined to be the limit, as r goes to infinity, of the proportion of elements in the ball of radius r which have the property P. We obtain a formula to compute the asymptotic densit…
Study GLS estimator properties in multivariate regression with heteroskedastic and autocorrelated errors.
problem Asymptotic properties of GLS estimator in multivariate regression with specific error structures.
method Derive Wald statistics for linear restrictions and assess their performance.
result Wald statistics remain robust to heteroskedasticity and autocorrelation.
CR method improves convergence for nonconvex optimization under KL property.
problem Improving convergence rate for nonconvex optimization problems.
method Cubic-regularized Newton's method exploiting Kurdyka-Lojasiewicz (KL) property.
result Asymptotic convergence rates of various optimality measures are fully characterized.
Study on finiteness property of right-angled Artin groups actions on extension graphs.
problem Finiteness property of hyperbolic simplicial actions on right-angled Artin groups.
method Analysis of right-angled Artin group actions on extension graphs, using asymptotic translation lengths and syllable lengths.
result Asymptotic translation lengths of elements in right-angled Artin groups are rational and have a common denominator under certain conditions.
Study maximum likelihood estimator for growth rate of jump-type CIR process.
problem Estimating growth rate of jump-type CIR process from continuous observations.
method Maximum likelihood approach, distinguishing subcritical, critical, and supercritical cases.
result Asymptotic properties including consistency and normality for different cases.
Study on large scale surjections and their properties.
problem Properties of large scale surjections and their invariants.
method Coarse disjointness and large scale n-to-1 functions.
result Characterization of asymptotic dimension for coarse structures.
The paper examines Adaptive Lasso and Transfer Lasso, highlighting their differences and proposing a new method.
problem Comparing and contrasting Adaptive Lasso and Transfer Lasso.
method Theoretical analysis of asymptotic properties and introduction of a new method.
result The Transfer Lasso method reduces non-asymptotic estimation errors compared to Adaptive Lasso.
The paper studies reward concentration in MDPs, covering asymptotic and non-asymptotic settings.
problem Reward concentration in Markov Decision Processes (MDPs).
method Unified approach to reward concentration in MDPs, including asymptotic and non-asymptotic bounds.
result Rate-equivalent definitions of regret for learning policies.
New insights into manifold properties using Seiberg-Witten and L2 harmonic theories.
problem Characterizing properties of 4-manifolds with specific geometric conditions.
method Combining Seiberg-Witten theory on compact manifolds and L2 harmonic theory on non-compact manifolds, with a new argument for asymptotic properties. result Found a pair of homeomorphic 4-manifolds with distinct geometric properties under Riemannian metrics.
We propose a novel time discretization for the log-normal SABR model and derive its asymptotic properties.
problem Analyzing the log-normal SABR model's time-discretized behavior and implied volatility surface.
method We use the Euler-Maruyama scheme for time discretization and derive asymptotic properties in the limit of large number of time steps.
result We derive an exact representation of the implied volatility surface for arbitrary maturity and strike in the asymptotic regime.
Paper addresses identifiability and asymptotics of ODE systems from noisy data.
problem Identifying parameters and causal structure of linear ODE systems from discrete observations.
method Developed sufficient conditions for identifiability, proved consistency and asymptotic normality of NLS estimator, constructed confidence sets, and inferred causal structure.
result Consistent and asymptotically normal parameter estimator for linear ODE systems under mild conditions.
Asymptotic behavior of energy of a harmonic map defined on an asymptotically hyperbolic manifold is considered. Using the growth of energy, we show that a harmonic map defined on some asymptotically hyperbolic manifolds has to be constant if the total energy is finite, or if the map approaches a point fast enough, in t…
The study shows mean partitions are consistent and asymptotically normal.
problem Lack of knowledge about the consistency of mean partitions in consensus clustering.
method Represented partitions as points in orbit space, used Fréchet means and stochastic programming, and analyzed continuous extensions of cluster criteria.
result Mean partitions are consistent and asymptotically normal under normal assumptions.
Metrics are semipositively curved if they meet a specific asymptotic condition.
problem Characterizing semipositively curved metrics in Hermitian geometry.
method Proving metrics are semipositively curved if and only if they satisfy an asymptotic extension property.
result Proves a specific condition for Griffiths semipositively curved metrics.
Extends flat submanifold properties from hyperbolic plane to symmetric spaces.
problem Spectral asymptotics for orbital integrals in symmetric spaces.
method Generalizes geodesic properties to maximal flat submanifolds.
result Establishes geometric properties of maximal flat submanifolds in symmetric spaces.
It is well-known that a paracompact space X is of covering dimension n if and only if any map f from X to a simplicial complex K can be pushed into its n-skeleton. We use the same idea to define dimension in the coarse category. It turns out the analog of maps f from X to K is related to asymptotically Lipschitz maps, …
We provide a detailed description of solutions of Curve Shortening in Rn that are invariant under some one-parameter symmetry group of the equation, paying particular attention to geometric properties of the curves, and the asymptotic properties of their ends. We find generalized helices, and a connection with curv…
Higher-dimensional Schwarzschild spacetimes violate the Penrose property.
problem Causal behavior of higher-dimensional Schwarzschild spacetimes.
method Analyzing causal properties in (2+1), (3+1), and (d+1) dimensions. result The Penrose property does not hold for (d+1) dimensional Schwarzschild if d>3.