Study shows how near crushing singularities, Kasner-like regions can exist.
problem Understanding spatial volume densities near crushing singularities.
method Relates existence of Kasner-like regions to asymptotics of spatial volume densities under scale-invariant curvature bounds.
result Kasner-like regions can exist near crushing singularities under certain curvature conditions.
The paper extends Hawking--Page solutions to various spacetimes with singularities.
problem Understanding the extensions of Hawking--Page solutions with different types of singularities.
method Kaluza--Klein reduction and Christodoulou's methods.
result Extensions of Lorentzian Hawking--Page solutions with null, spacelike singularities, and Cauchy horizons of Taub--NUT type are proven.
A trace on the C^*-algebra A of quasi-local operators on an open manifold is described, based on the results in \cite{RoeOpen}. It allows a description `a la Novikov-Shubin \cite{NS2} of the low frequency behavior of the Laplace-Beltrami operator. The 0-th Novikov-Shubin invariant defined in terms of such a trace is pr…
In this paper we are interested in term structure models for pricing zero coupon bonds under rapidly oscillating stochastic volatility. We analyze solutions to the generalized Cox-Ingersoll-Ross two factors model describing clustering of interest rate volatilities. The main goal is to derive an asymptotic expansion of …
An almost Einstein manifold satisfies equations which are a slight weakening of the Einstein equations; Einstein metrics, Poincare-Einstein metrics, and compactifications of certain Ricci-flat asymptotically locally Euclidean structures are special cases. The governing equation is a conformally invariant overdetermined…
Researchers create finite time blow-up solutions for harmonic map flow into S^2.
problem Constructing finite time blow-up solutions for harmonic map flow into S^2.
method Constructing finite time blow-up solutions using asymptotically singular scaling and reverse bubbling.
result Finite time blow-up solutions constructed precisely at given points in the domain.
Study asymptotic behaviors of solutions near singular boundaries for the Yamabe problem.
problem Boundary behavior of the singular Yamabe problem near singular boundaries.
method Analysis of asymptotic behaviors and derivation of optimal estimates for background metrics.
result Solutions are well approximated by solutions in tangent cones at singular points.
Study finds solutions to Yamabe equation with specific behavior near singular points.
problem Existence of solutions with prescribed asymptotic behavior near singular points of the Yamabe equation.
method Analysis of positive solutions with isolated singularities and asymptotic expansions.
result Existence of solutions with arbitrarily high order of approximation near singular points.
Polyhomogeneous expansions for Calabi-Yau metrics near singularities.
problem Analyzing metrics near conical singularities of Calabi-Yau conifolds.
method Weighted Melrose-type blow-ups, gluing, and solving complex Monge-Ampère equations.
result Polyhomogeneous expansions of smooth Calabi-Yau metrics on resolutions and smoothings.
Study heat trace expansion on manifolds with conic points.
problem Analyzing heat diffusion on manifolds with sharp corners.
method Using the Singular Asymptotics Lemma to derive an expansion.
result Detailed asymptotic expansion reveals geometric insights.
Ricci flow modelled on specific singularities on closed manifolds.
problem Analyzing singularities in Ricci flows.
method Closed manifold Ricci flow with singularity modeled on asymptotically conical shrinkers.
result Ricci flow solution forms a singularity that matches the given asymptotically conical shrinker.
Study of Lagrangian mean curvature flow with equivariant symmetry.
problem Understanding singularities in Lagrangian mean curvature flow.
method Structural theorems about blowups of finite-time singularities.
result Classification of singularities in equivariant case.
Positive mass theorem for singular hyperbolic spaces proved.
problem Positive mass theorem for singular asymptotically hyperbolic manifolds.
method Careful analysis of the Yamabe equation.
result Positive mass theorem established for singular asymptotically hyperbolic manifolds.
Uniform elliptic theory for Dirac operators on orbifold resolutions.
problem Analyzing Dirac operators on orbifold resolutions.
method Viewing orbifolds as conically fibred singular spaces and resolving them by gluing asymptotically conical fibrations.
result Uniform index formula for Dirac operators on orbifold resolutions.
Develops local theory for singular spacetimes becoming asymptotically self-similar.
problem Construction of singular spacetimes in all dimensions.
method Local theory and construction of exact self-similar solutions.
result Construction of exact self-similar solutions corresponding to formal asymptotic expansions.
Backwards uniqueness proved for flows with asymptotically conical singularities.
problem Proving uniqueness of mean curvature flows with specific singularities.
method Developed new global tools to handle singularities, asymptotic structure, and smooth parts of flows.
result Backwards uniqueness for mean curvature flows with asymptotically conical singularities proved.
Proves positive mass theorem for AF spin manifolds with conical singularities.
problem Proving the positive mass theorem for singular metrics on AF manifolds.
method Analyzes AF spin manifolds with isolated conical singularities, allowing topological singularities.
result Proves the positive mass theorem for AF spin manifolds with conical singularities.
Study of bubble-sheet singularity in mean curvature flow.
problem Analyzing a specific type of singularity in mean curvature flow.
method Deriving an asymptotic profile for a neighborhood of the singularity.
result An asymptotic profile for a neighborhood of the singularity is derived.
The paper establishes criteria for spacetime inextendibility using asymptotic volume-distance-ratio analysis.
problem Determining inextendibility of spacetimes near singularities.
method Asymptotic analysis of volume-distance-ratio (VDR) to prove inextendibility criteria.
result Failure of VDR convergence to the Minkowski value implies inextendibility of spacetime.
Prove Riemannian positive mass theorem for asymptotically flat metrics with low-codimension singularities.
problem Prove Riemannian positive mass theorem for asymptotically flat metrics with low-codimension singularities.
method Prove Riemannian positive mass theorem for asymptotically flat metrics with low-codimension singularities.
result Prove Riemannian positive mass theorem for asymptotically flat metrics with low-codimension singularities.
Uniqueness of conical flows helps understand singularities in surface flows.
problem Understanding singularities in surface flows.
method Analyzing asymptotically conical tangent flows.
result Uniqueness of multiplicity-one asymptotically conical tangent flows.
Study precise asymptotic behavior of functions in singular metric spaces.
problem Singular metric spaces with incomplete geometry.
method Expansions of quasi-harmonic and eigenfunctions.
result More precise description of asymptotic behavior at infinity.
Proves mass theorem for AF manifolds with conical singularities.
problem Proving the positive mass theorem for specific types of manifolds.
method Conformal blow up technique applied to AF manifolds with isolated conical singularities.
result Positive mass theorem proven for the specified manifolds.
The paper studies a flow of Legendre curves, generalizing the inverse curvature flow of regular curves.
problem Analyzing the inverse curvature flow of Legendre curves.
method Investigates the unique existence, monotonicity, and asymptotic behavior of the flow.
result The flow asymptotically converges to a self-similar solution, categorized by initial curve.
A singularity theorem based on asymptotic volume growth
problem Proving singularity theorems
method Introducing asymptotic volume-expansion invariants
result Proving an explicit upper bound on the time-separation from a hypersurface to its chronological past
Desingularizes conically singular Cayley submanifolds.
problem Constructing fibrations of compact Spin(7) manifolds by Cayley submanifolds.
method Desingularization through gluing rescaled asymptotically conical submanifolds.
result Conically singular Cayley submanifolds can be desingularized.
Optimizes portfolios in fast mean-reverting markets, achieving asymptotic efficiency.
problem Optimizing portfolios in markets with fast mean-reverting returns and volatility.
method Proposes a zeroth order strategy and uses singular perturbation method for asymptotic optimality.
result Shows asymptotic optimality of the proposed strategy under specific assumptions.
This paper analyzes portfolio optimization with multi-scale volatility.
problem Optimizing portfolio under multi-scale volatility in a stochastic environment.
method Zeroth-order strategy followed by first-order approximation via PDE analysis.
result Asymptotic optimality of the proposed strategy in specific families of controls.
SLT explains neural network success by closing theory-practice gap.
problem Failure of classical inference and learning theory in modern neural networks.
method Physics-inspired Singular Learning Theory (SLT) applied to neural networks.
result SLT recovers known and novel scaling laws for neural network phase transitions.
Study oscillatory integrals with degenerate singular points in multivariable phase functions.
problem Analyzing oscillatory integrals with degenerate singular points in phase functions.
method Using asymptotic expansions and results from one variable, the study examines multivariable phase functions.
result Asymptotic expansions of oscillatory integrals for multivariable phase functions with degenerate singular points.
We prove that the asymptotic completion of a developable Möbius strip in Euclidean three-space must have at least one singular point other than cuspidal edge singularities. Moreover, if the strip contains a closed geodesic, then the number of such singular points is at least three. These lower bounds are both sharp.
Study on solutions near isolated singularities in 6D Yamabe equation.
problem Behavior of solutions near isolated singularities in 6D Yamabe equation.
method Analyze asymptotic behavior of local solutions in non-conformally flat metrics.
result Solutions are asymptotically close to Fowler solutions in 6D.
The aim of this paper is to prove the existence of weak solutions to the equation Δu+up=0 which are positive in a domain Ω⊂RN, vanish at the boundary, and have prescribed isolated singularities. The exponent p is required to lie in the interval (N/(N−2),(N+2)/(N−2)). We also prove the exist…
We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold. This extends the classical notion of asymptotic directions usually defined on smooth submanifolds. We get a simple expression of these cones for polyhedra in E^3, as well as converge…
We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, Δu + n(n-2)/4 u^{(n+2)(n-2) = 0, in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, by Caffarelli, Gidas a…
We consider stochastic control systems affected by a fast mean reverting volatility Y(t) driven by a pure jump Lévy process. Motivated by a large literature on financial models, we assume that Y(t) evolves at a faster time scale εt than the assets, and we study the asymptotics as $\varepsilon\t…
Paper analyzes singular subspace estimation in noisy matrix models.
problem Estimating low-rank signals in noisy matrix data.
method Asymptotic distributional theory, extreme value theory, saddle point approximation, random matrix theory.
result Plug-in test statistic based on two-to-infinity norm has higher power for detecting structured alternatives.
Researchers found a family of Sp(2)-invariant solitons for Laplacian flow.
problem Analyzing Sp(2)-invariant solitons for Laplacian flow on the 4-sphere.
method Used mathematical analysis and asymptotic cone determination.
result Identified a 1-parameter family of Sp(2)-invariant expanding solitons with specific asymptotic behavior.
Study conic singular manifolds, proving Lipschitz normal embedding.
problem Understanding metric properties of conic singular manifolds.
method Analyzing interplay between conic and asymptotically conic behavior.
result Proves Lipschitz normal embedding for conic singular sub-manifolds.
Study of mean curvature flows with conical singularities using mathematical techniques.
problem Understanding the dynamics of mean curvature flows near conical singularities.
method Feynman-Kac formula and invariant cone method for noncompact settings.
result Generic initial perturbations avoid conical singularities in mean curvature flows.
Study of solitons in Laplacian flow on 7-manifolds.
problem Understanding finite-time singularities of Laplacian flow on G_2-structures.
method Systematic study of cohomogeneity-one solitons with Sp(2) and SU(3) symmetry groups.
result Existence and classification of smoothly-closing solitons, including complete and incomplete solutions.
The paper derives an equation linking WAIC and WBIC for singular models.
problem In singular models, conventional criteria fail due to likelihood and posterior breakdown.
method Theoretical derivation linking WAIC and WBIC.
result An asymptotic equation linking WAIC and WBIC for singular models.
Study neckpinch singularities in Ricci flow with cylindrical symmetry.
problem Understanding the asymptotic behavior of neckpinch singularities in Ricci flow.
method Rigorous analysis under Type-I assumption for general symmetric initial data.
result Previously constructed asymptotic profiles are the only possibilities.
Heat trace analysis reveals singularity properties on wedge-shaped manifolds.
problem Analyzing heat trace asymptotics on manifolds with wedge-like singularities.
method Heat trace expansion and asymptotic analysis on manifolds with wedge-like singularities.
result Two terms in the heat trace expansion contain information about the singularity.
Paper solves complex equations near cone points.
problem Solving complex Monge-Ampère equations with conical singularities.
method Proved asymptotic expansion of solutions.
result Found solutions to conical Kähler-Einstein metrics.
Study on naked singularities without symmetry, forming incomplete future null infinity and singular inner Cauchy horizon.
problem Formation of naked singularities in Einstein-scalar field system without symmetry assumptions.
method Employing four-type differences and scale-invariant weighted norms to control geometry.
result Global naked singularity structure with incomplete future null infinity and singular inner Cauchy horizon.
Developed theory of Perelman's W-functional on manifolds with conical singularities.
problem Analyzing manifolds with conical singularities using Perelman's W-functional.
method Theory development and mathematical analysis on manifolds with isolated conical singularities.
result Existence and asymptotic order of minimizers for the W-functional on manifolds with conical singularities.
We simplify swallowtail singularity and classify nearby curves.
problem Understanding swallowtail singularity and its nearby curves.
method Coordinate transformation on source and isometry on target.
result Classification of asymptotic and characteristic curves near swallowtail.