The paper proves a unique Kollár component is K-semistable among plt blow ups of a klt singularity.
problem Stability of valuations and Kollár components in klt singularities.
method Proving minimization of the normalized volume function yields K-semistable Kollár components.
result There is at most one Kollár component that is (log-)K-semistable among plt blow ups of a klt singularity.
The paper proves ACC for local volumes under boundedness conditions.
problem Proving the ACC conjecture for local volumes of klt singularities.
method Analyzing klt singularities with bounded ambient germs.
result ACC conjecture for local volumes holds under bounded conditions.
Study uniform K-stability and its connection to log Fano pairs' plt blowups.
problem Testing uniform K-stability of log Fano pairs.
method Evaluate volume function invariants for all plt blowups.
result Establishes uniform K-stability criteria for log Fano pairs.
The paper tackles the computational complexity of PLT algorithms, providing tractable solutions.
problem Finding a tree structure that results in a PLT with low computational costs and low statistical error.
method The paper shows that finding an optimal tree structure is NP-complete. It provides linear time solutions for specific cases and an O(logm) approximation for the general case. result The paper provides tractable solutions for finding a tree structure with low computational costs and low statistical error.
Paper improves XMLC by no-regretly generalizing HSM to PLTs.
problem Efficiently tagging instances with a small subset of relevant labels from a large pool.
method Probabilistic label trees (PLTs) as a no-regret multi-label generalization of hierarchical softmax (HSM).
result XT (extremeText) outperforms HSM with pick-one-label heuristic and XML-CNN.
Probabilistic label trees improve XMLC by organizing labels hierarchically.
problem Efficiently tagging instances with a small subset of relevant labels from a large pool.
method Introduce and analyze probabilistic label trees (PLTs) as a generalization of hierarchical softmax for multi-label problems.
result PLTs are consistent for various performance metrics and can be trained online without prior knowledge.
Study blow-ups of locally conformal symplectic manifolds.
problem Blow-ups of locally conformal symplectic manifolds.
method Show existence of locally conformal symplectic structure on blow-ups.
result Existence of locally conformal symplectic structure on blow-ups.
Study blow-ups in generalized complex geometry using holomorphic ideals.
problem Blow-ups in generalized complex geometry.
method Introduce holomorphic ideal to define blow-ups in smooth manifolds. Identify suitable submanifolds and provide conditions for blow-ups.
result Necessary and sufficient conditions for generalized Poisson submanifolds to carry a canonical holomorphic ideal and for blow-ups to be generalized complex.
Study on half spheres solves Nirenberg problem with complex blow-up analysis.
problem Prescribing scalar curvature on half spheres.
method Refined blow-up analysis of finite energy approximated solutions.
result Complex blow-up points and vortex problems reveal new connections.
The paper defines and constructs almost complex blow-ups on 4D almost complex manifolds.
problem Existence and uniqueness of almost complex blow-ups on almost complex manifolds.
method Definition and construction of almost complex blow-ups, proving their existence and uniqueness.
result Existence and uniqueness of almost complex blow-ups on 4D almost complex manifolds.
Formula derived for holomorphic Poisson blow-ups.
problem Invariance of Koszul-Brylinski homology under Poisson blow-ups.
method Blow-up formula derivation for holomorphic Koszul-Brylinski homologies.
result Invariance of E1-degeneracy of Dolbeault-Koszul-Brylinski spectral sequence. Study identifies numerical signs of blow-up in hydrodynamic equations.
problem Determining if numerical results of blow-up are genuine or artifacts.
method Geometrically consistent spatiotemporal discretization of complexified Euler equations.
result Identification of a signature based on supremum norm growth rates of vorticity.
Extends cutting and blowing up to nonrational symplectic settings.
problem Nonrational symplectic toric structures.
method Cutting and blowing up in nonrational directions.
result Extension to symplectic toric manifolds and orbifolds.
Formula derived for Bott-Chern classes in complex blow-ups.
problem Calculating Bott-Chern classes in blow-ups of complex manifolds.
method Proved blow-up formula for Bott-Chern classes, established Riemann-Roch without denominators.
result Formula for Bott-Chern classes in blow-ups.
Explains blow-ups for Lie groupoids and algebroids, comparing different methods.
problem Blow-ups of Lie groupoids and algebroids.
method Detailed explanation of various blow-up constructions.
result Different blow-up constructions for Lie groupoids and algebroids are shown to fit into a general geometric framework.
The paper derives a formula for Chow weights of toric blow-ups.
problem Chow weights of toric blow-ups.
method Combinatorial formula derived from toric manifold and Delzant polytope.
result Explicit formula for Chow weights of blow-ups.
Study of blow-ups in generalized Kähler geometry with specific conditions.
problem Blow-ups in generalized Kähler manifolds with generalized Poisson submanifolds.
method Blow-up theory for generalized Kähler manifolds, lifting bi-Hermitian structure, deformation procedure based on potentials.
result Degenerate bi-Hermitian structure on blow-up can be deformed into non-degenerate structure.
New proof of blow-up formula for Morse-Novikov cohomology.
problem Blow-up formula for Morse-Novikov cohomology.
method Introducing relative Morse-Novikov cohomology and using sheaf cohomology.
result Explicit isomorphism in relative Morse-Novikov cohomology.
Study on curvature blow-up rates in black hole interiors from gravitational collapse.
problem Understanding curvature blow-up rates in black hole interiors during gravitational collapse.
method Investigation of spherically symmetric Einstein-scalar field spacetimes, focusing on blow-up rates of curvature and mass.
result Kretschmann scalar blows up faster than in Schwarzschild setting, indicating a new blow-up phenomenon.
Computes cohomologies of blow-ups and projective bundles.
problem Cohomologies of blow-ups and projective bundles.
method Computes double complex of differential forms on projective bundles and blow-ups.
result Formulas for all cohomologies associated with the complex.
The paper examines the blow-up of Ricci curvatures in conformal metrics.
problem Characterizing the blow-up set of Ricci curvatures in conformal metrics.
method Analyzing the blow-up phenomena of Ricci curvatures on domains close to a limit set of lower dimension.
result Characterization of the blow-up set according to the Yamabe invariant of the manifold.
We show that the blow-up of a generalized Kahler 4-manifold in a nondegenerate complex point admits a generalized Kahler metric. As with the blow-up of complex surfaces, this metric may be chosen to coincide with the original outside a tubular neighbourhood of the exceptional divisor. To accomplish this, we develop a b…
This paper studies a specific blow-up algorithm for sop polynomials and their RLCT.
problem Determining the RLCT of sum-of-products polynomials through blow-up.
method Investigates a specific blow-up algorithm for sop polynomials to resolve their singularities.
result It is possible to resolve the singularities of sop polynomials using a specific blow-up algorithm.
The abstract proves a Leray-Hirsch theorem and blow-up formula for Dolbeault cohomology.
problem Cohomology on complex manifolds, especially non-compact ones.
method Proves a Leray-Hirsch theorem and an explicit blow-up formula.
result Explicit blow-up formula for Dolbeault cohomology on complex manifolds.
Solutions blow up for Yamabe problem on umbilic manifolds with nonzero Weyl tensor.
problem Yamabe problem on manifolds with umbilic boundary
method Building blowing-up solutions
result Solutions blow up for linear perturbation of Yamabe problem
The paper analyzes high-dimensional sphere solutions to the Nirenberg problem with residual mass.
problem The Nirenberg problem on high-dimensional spheres with residual mass.
method Analysis of subcritical approximations and blowing up solutions.
result Comprehensive description of blowing up solutions, including blow-up points and rates.
The Yamabe flow can blow up in infinite time with small perturbations.
problem Understanding the behavior of the Yamabe flow under small perturbations.
method Constructive proof using solutions of the Yamabe problem on the unit sphere as blow-up profiles.
result The Yamabe flow can blow up at multiple points on a Riemannian manifold in infinite time with small perturbations.
The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up …
Study on spinor field equation on spheres, focusing on blow-up analysis.
problem Spinorial Yamabe problem on spheres.
method Variational methods, blow-up analysis.
result Blow-up profile for the spinorial Yamabe type equation on Sm. Paper analyzes harmonic maps blow-up neck regions.
problem Understanding blow-up regions in harmonic maps.
method Proved refined estimates in neck regions.
result New estimate reveals neck shape and inequality about nullity and index.
Study solutions to metric equations on modified surfaces.
problem Finding constant scalar curvature Kähler metrics on modified surfaces.
method Expanding solutions to extremal metric type equations.
result Developed methods to solve metric equations on modified surfaces.
We prove a gluing formula for Seiberg--Witten invariants which describes in particular the behaviour of the invariant under blow-up and rational blow-down.
Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.
problem Investigate the homotopy of blow ups in algebraic and symplectic geometry.
method Develops fibrewise surgery theory and a purely homotopy theoretic approach.
result Obtained homotopy decompositions of the based loop space on blow ups.
New extremal Poincaré type metrics found via blow-up theorem.
problem Finding new extremal Poincaré type metrics.
method Proved Arezzo-Pacard-Singer blow-up theorem for Poincaré type metrics and applied to new examples.
result Found new examples of extremal Poincaré type metrics with an additional obstruction.
Study on blow-up behavior of sign-changing solutions for Yamabe equation.
problem Blow-up behavior of sign-changing solutions for Yamabe equation.
method Construction of a smooth metric on space forms to prove blow-up at lowest energy level.
result Blow-up occurs at the lowest energy level for sign-changing solutions in dimensions 11 to 24.
Proves inextendibility of weak null singularities from curvature blow-up.
problem Inextendibility of weak null singularities in the context of curvature blow-up.
method Introduces a new strategy to infer Cloc0,1-inextendibility from curvature blow-up. result Expected to contribute to the resolution of strong cosmic censorship conjecture.
Study on solutions of conformal equations, proving bounds and profiles.
problem Analyzing solutions of conformally invariant equations on Euclidean domains.
method Established blow-up profiles and heights of solutions around blow-up points.
result Proved bounds on distances and heights of solutions around blow-up points.
Desingularizes singular foliations with a locally compact groupoid.
problem Handling singularities in foliations.
method Blow-up construction of smooth manifolds and groupoids.
result Locally compact locally Hausdorff groupoid desingularizes singular foliations.
Corrected Monti's blow-up analysis for H-minimizing sets in Heisenberg group.
problem Blow-up analysis of H-minimizing sets in Heisenberg group with corrected partial differential equation.
method Revised Monti's results on blow-ups of H-perimeter minimizing sets in Hn and corrected the partial differential equation for the limit function. result Corrected the partial differential equation for the limit function of blow-ups in Heisenberg group.
Ancient solutions in Lagrangian flow are classified based on their blow-down.
problem Understanding ancient solutions in Lagrangian mean curvature flow.
method Structural and classification results for ancient solutions, focusing on the almost calibrated case.
result Classification of Type II blow-ups in terms of their blow-down.
The article proves conditions for blow-ups of lcK spaces to remain lcK.
problem Conditions for blow-ups of locally irreducible lcK spaces to remain lcK.
method Proves conditions for blow-ups of lcK spaces to remain lcK.
result Blow-ups of locally irreducible lcK spaces are lcK if and only if the original space is induced gcK.
Formula for Dolbeault cohomology groups on complex manifolds with vector bundles.
problem Calculating Dolbeault cohomology groups on complex manifolds with vector bundles.
method Sheaf-theoretic approach to derive a blow-up formula.
result Uniform presentation of vanishing theorems and blow-up invariance of holomorphic invariants.
Finite-time blow-up in Yang-Mills flow for small energy initial connections.
problem Finite-time blow-up of Yang-Mills flow solutions.
method Analyzing the Yang-Mills flow on Riemannian and Kähler manifolds.
result Finite-time blow-up occurs for small energy initial connections.
The study examines blow-up and global existence of solutions for porous medium equation on curved manifolds.
problem Analyzing blow-up and global existence of solutions for porous medium equation on negatively curved manifolds.
method Examined the porous medium equation with power-type reaction terms on negatively curved Riemannian manifolds.
result For p>m, small data give global solutions; for p<m, large data blow up in infinite time. Study shows continuity and geometric regularity of Kähler-Ricci flow blow-up limits.
problem Geometric regularity of blow-up limits of the Kähler-Ricci flow.
method Established geometric regularity for Type I blow-up limits based on sequences of Ricci vertices.
result The limiting flow is continuous in time in Gromov-Hausdorff and Gromov-W1 distance. We show that at generic points blow-ups/tangents of differentiability spaces are still differentiability spaces; this implies that an analytic condition introduced by Keith as an inequality (and later proved to actually be an equality) passes to tangents. As an application, we characterize the p-weak gradient on iter…
Study of curvature blow-up in noncompact hypersurfaces using mean curvature flow.
problem Curvature blow-up in mean curvature flow of noncompact hypersurfaces.
method Rotationally symmetric solutions constructed with precise asymptotics.
result Highest curvature concentrates at the tip and blows up at rate (T−t)−1. CR structure on S³ with non-compact solutions to CR Yamabe problem.
problem Existence of non-compact solutions to CR Yamabe problem.
method Deforming standard CR structure of S³, using Lyapunov-Schmidt method.
result Existence of a blowing-up sequence of solutions.