The study proves regularity of metrics on complex domains.
problem Regularity of metrics on complex domains.
method Abstract complex manifold theorem with Monge-Ampère exhaustions of Ck regularity. result Existence of bounded open neighborhoods with smooth metrics.
By a theorem of Greene and Wu, a noncompact connected Riemannian manifold admits a smooth strictly subharmonic exhaustion function. Demailly provided an elementary proof of this fact. A further simplification of Demailly's proof and some (mostly known) applications are described. Applications include the fact that the …
The paper explores different smooth map notions on convex sets and their relationships.
problem Exploring and comparing different smooth map notions on convex sets.
method Constructing a function that doesn't extend to a smooth function on any open neighborhood but does for Ck functions. result Diffeological and Sikorski smoothness notions do not coincide for all convex sets.
The paper studies Kähler metrics from finite Monge-Ampère mass exhaustion functions.
problem Investigating the spectrum of complete Kähler metrics from finite Monge-Ampère mass exhaustion functions.
method Analyzing logarithmic potentials and the associated complete Kähler metrics, proving bounds on the spectrum using the finite Monge-Ampère mass condition.
result The lower bound of the spectrum of the Laplace-Beltrami operator is n2 under the finite Monge-Ampère mass condition. The paper explores symplectic foliations and their leaves on manifolds.
problem Which manifolds can be realized as leaves of codimension-1 symplectic foliations?
method Observations and deformations of symplectic structures; examples of manifolds.
result Examples of manifolds that can be realized as leaves but not as symplectic leaves.
The paper explores curvature-free effects in manifolds with volume growth and ends-counting.
problem Investigating curvature-free effects in manifolds with volume growth and ends-counting.
method Establishing two main theorems about volume growth and ends-counting.
result Proves the existence of smooth bounded mean-concave exhaustion and escaping geodesic lines.
Survey on preserving curvature bounds for non-smooth Ricci flow.
problem Preserving curvature bounds for non-smooth initial data in Ricci flow.
method Survey of various weak initial data and preservation of curvature bounds.
result Various curvature lower bounds preserved up to a constant for non-smooth initial data.
A two-step approach efficiently selects hyperparameters for FCMs.
problem Efficiently selecting hyperparameters for FCMs in a computationally expensive process.
method Two-step sequential approach: first estimate context length k, then estimate α.
result The proposed method achieves comparable compression performance to exhaustive search but with reduced computational cost.
Typical existence result on Ricci-flat metrics is in manifolds of finite geometry, that is, on F=Fˉ−D where Fˉ is a compact Kähler manifold and D is a smooth divisor. We view this existence problem from a different perspective. For a given complex manifold X, we take a suitable exhaustion $\{X_r\}_{r>0}…
Study exhausts curve complex on nonorientable surfaces using finite rigid sets.
problem Exhausting curve complexes on nonorientable surfaces.
method Using finite rigid sets to exhaust curve complex.
result Improves exhaustion result for (g,n)=(3,0) or g+n≥5. For nonorientable surfaces, curve complexes can be exhausted by finite superrigid sets.
problem Exhausting curve complexes on nonorientable surfaces.
method Using finite superrigid sets for exhaustion.
result An exhaustion of curve complexes by finite superrigid sets for (g,n)eq(1,2) and g+neq4. This paper exhausts curve complexes on non-orientable surfaces.
problem Proving exhaustion of curve complexes on non-orientable surfaces.
method Proving exhaustion via rigid expansions and graph endomorphisms.
result Any graph endomorphism of curve complexes whose restriction to a finite rigid set is injective is induced by a homeomorphism.
This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic functi…
Proposes methods for selecting sparse variables in linear regression.
problem Selecting sparse variables in linear regression models.
method K-sparse exhaustive search (ES-K) and K-sparse approximate exhaustive search (AES-K) methods.
result AES-K method effectively reconstructs density of states for large problems.
Probabilistic model for exhaustion in infinite-genus curve complexes.
problem Action rigidity in infinite-genus curve complexes.
method Costa and Farber's model for random simplicial complexes.
result Probabilistic evidence for exhaustion via rigid expansions.
Proves existence of functions for Ricci flow and Yau's conjecture.
problem Existence of exhaustion functions for Ricci flow and Yau's conjecture.
method Adapting L.-F. Tam's method for proving existence.
result Short-time existence of Ricci flow and uniformization of surfaces.
grangersearch tests causal relationships in time series data.
problem Testing causal relationships between multiple time series.
method Exhaustive pairwise search, automatic lag order optimization, tidyverse integration.
result Automated Granger causality testing simplifies causal analysis.
Alexander self-dual complexes yield smooth compactifications of M0,n with explicit Chow rings.
problem Understanding the structure of moduli spaces M0,n using Alexander self-dual complexes. method Explicit description of Chow rings, computation of Chern classes and intersection numbers.
result Derivation of a recursion for intersection numbers in ASD compactifications.
Researchers solved a model of an exhaustible resource with stochastic discoveries.
problem Optimal exploration of an exhaustible resource with uncertain discoveries.
method Impulse control and Poisson process of new discoveries.
result A frontier of critical levels of proven reserves exists, above which exploration is stopped.
Let S be a connected orientable surface of finite topological type. We prove that there is an exhaustion of the curve complex C(S) by a sequence of finite rigid sets.
Study exhaustions for complex orbits in almost homogeneous manifolds.
problem Complex orbits in almost homogeneous manifolds.
method Complex homogeneous Monge-Ampère equations.
result Rigidity results on complex spaces.
Researchers exhaustively cover pants graphs of spheres with punctures using finite rigid sets.
problem Covering all pants graphs of punctured spheres efficiently.
method Constructing a sequence of finite rigid sets in the pants graph.
result The union of these sets covers the entire pants graph of the punctured sphere.
We give two applications of the the duality between the complex Homogeneous Monge-Ampère Equation (HMAE) and the Hele-Shaw flow. First, we prove existence of smooth boundary data for which the weak solution to the Dirichlet problem for the HMAE over P1×D is not twice differentiable a…
Constructs improperly embedded minimal disks with curvature issues.
problem Properly embedded minimal disks with curvature issues in a cylinder.
method Constructs a sequence of improperly embedded minimal disks with curvature blow-up at a single point.
result The disks are not properly embedded in open subsets of R^3 that exhaust R^3.
Researchers exhaust curve graph using rigid expansions on surfaces.
problem Exhausting the curve graph of surfaces with genus ≥ 3.
method Constructing a finite set of curves and using iterated rigid expansions.
result The constructed set exhausts the curve graph via rigid expansions.
The paper proves isoperimetric regions in specific manifolds cannot escape to infinity.
problem Exploring isoperimetric regions in asymptotically hyperbolic manifolds.
method Analyzing scalar curvature and Hawking mass to prove region exhaustivity.
result Isoperimetric regions with scalar curvature ≥ -6 cannot escape to infinity.
Bayesian model adapts to changing classes in machine learning.
problem Non-stationary environments with incomplete class information.
method Doubly non-parametric Bayesian Gaussian mixture model.
result Model can grow to accommodate any number of classes.
The paper explores uniform perfectness and centers in Morse boundaries.
problem Detecting κ-center exhaustivity in uniformly perfect Morse boundaries. method Analyzes CAT(0) and geodesic spaces, using visual boundary data and metric transforms.
result Fixed-basepoint uniform perfectness is insufficient for κ-center exhaustivity. Proposes a taxonomy for economic policies.
problem Lack of a standardized list of economic policies.
method Develops a tree taxonomy to categorize economic policies.
result Constructs an exhaustive list of economic policies.
Regularity properties of intrinsic objects for a large class of Stein Manifolds, namely of Monge-Ampère exhaustions and Kobayashi distance, is interpreted in terms of modular data. The results lead to a construction of an infinite dimensional family of convex domains with squared Kobayashi distance of prescribed regula…
Paper proposes using pairwise feature comparisons to infer modification costs for user recourse.
problem Learning and inferring user preferences for modifying features in black-box models.
method Bradley-Terry model for inferring feature-wise costs from non-exhaustive human comparison surveys.
result Non-exhaustive human surveys can efficiently learn feature costs, enabling recourse finding.
KOLMOGOROV-OPTIMAL RESOLUTION ESTIMATION (KORE) solves spline regression without exhaustive search
problem Hyperparameter tuning in spline regression
method Solving for optimal resolution analytically
result KORE matches exhaustive cross-validation and outperforms tuned models
Neurally-Guided Structure Inference combines search and data-driven methods for efficient, robust structure inference.
problem Combining the advantages of exhaustive search and data-driven methods for structure inference.
method Neurally-Guided Structure Inference (NG-SI) uses a neural network to guide hierarchical search over structures.
result NG-SI outperforms search-based and data-driven methods on probabilistic matrix decomposition and symbolic program parsing.
We study infinite-type 3-manifolds with compact hyperbolizable pieces.
problem Characterizing infinite-type 3-manifolds with hyperbolizable pieces.
method Compact exhaustion and topological conditions for hyperbolic metrics.
result Necessary and sufficient conditions for complete hyperbolic metrics.
In this paper, we prove that the L^2 Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, under some hypotheses. We also prove that some L^2 spectral invariants can be approximated by the corresponding average spectral invariants of a regular exhaustion. …
Study ancient solutions to free boundary mean curvature flow in convex manifolds.
problem Understanding ancient solutions to free boundary mean curvature flow in convex manifolds.
method Establish rigidity results and construct foliations to describe ancient solutions.
result Ancient solutions to free boundary mean curvature flow are rigid and exhaust all possibilities under certain conditions.
Generalizes Nakano-positivity to Hilbert space fields.
problem Extending Nakano-positivity to Hilbert space fields.
method Exhaustion arguments to generalize the theorem.
result Log-plurisubharmonic variation results for Stein manifolds.
A compact real analytic Riemannian manifold M admits a canonical complexification with plurisubharmonic exhaustion function satisfying the homogeneous complex Monge-Ampere equation, called a Grauert tube. From the point of view of complex analysis, several authors have considered whether a given complex manifold can ar…
A finitely presented group is weakly geometrically simply connected (wgsc) if it is the fundamental group of some compact polyhedron whose universal covering is wgsc i.e. it has an exhaustion by compact connected and simply connected sub-polyhedra. We show that this condition is almost-equivalent to Brick's qsf propert…
Constructs uniformly positive scalar curvature metrics on open manifolds
problem Finding uniformly positive scalar curvature metrics on open manifolds
method Using Morse functions and exhaustion
result Proving the existence of uniformly positive scalar curvature metrics
New research shows IBM's GDX algorithm outperforms Vytelingum's Adaptive-Aggressive strategy in market simulations.
problem Comparing the performance of adaptive-aggressive trading algorithms in various market scenarios.
method Exhaustive testing across a wide range of market environments using large-scale compute facilities.
result Vytelingum's Adaptive-Aggressive strategy is consistently outperformed by IBM's GDX algorithm in simple market conditions.
In this paper we study the Sasakian geometry on S^3-bundles over a Riemann surface of genus g>0 with emphasis on extremal Sasaki metrics. We prove the existence of a countably infinite number of inequivalent contact structures on the total space of such bundles that admit 2-dimensional Sasaki cones each with a Sasaki m…
We generalize A. Borbély's condition for the conclusion of the Omori-Yau maximum principle for the Laplace operator on a complete Riemannian manifold to a second-order linear semi-elliptic operator L with bounded coefficients and no zeroth order term. Also, we consider a new sufficient condition for the existence of …
Following a purely algebraic procedure, we provide an exhaustive classification of local Weyl-invariant scalar densities in dimension D=8.
The paper explores a generalized notion of transversality in harmonic analysis.
problem Generalizing transversality to collections of submanifolds in harmonic analysis.
method Examining three related concepts of transversality in harmonic analysis and showing their equivalence.
result Three concepts of transversality in harmonic analysis are equivalent.
We prove that if a family of metrics, gi, on a compact Riemannian manifold, Mn, have a uniform lower Ricci curvature bound and converge to g∞ smoothly away from a singular set, S, with Hausdorff measure, Hn−1(S)=0, and if there exists connected precompact exhaustion, Wj, of Mn∖S s…
Interactive framework improves understanding of deep neural networks.
problem False sense of comprehension from static explanation methods.
method Interactive framework allowing exhaustive inspection and testing of decisions.
result Interactive approach leads to better understanding of complex decision boundaries.
Pattern sampling reduces time series classification complexity.
problem High computational complexity of exhaustive search for shapelets.
method Pattern sampling using a weighted trie to extract discriminative patterns.
result Significant reduction in computational and memory resources.