Refined theorem on linear perturbations with applications in singularity theory and optimization.
problem Linear perturbations and their implications in singularity theory and optimization.
method New perspective of Hausdorff measures for refined transversality theorem.
result Applications in singularity theory and optimization.
New Alexander polynomial for singular knots improves upon existing methods.
problem Defining a polynomial invariant for singular knots.
method Introducing a perturbed Alexander polynomial.
result The new polynomial agrees with previous definitions for long knots.
New framework for higher-order singular-value derivatives of rectangular matrices.
problem Challenging to derive higher-order Fréchet derivatives of singular values in real rectangular matrices.
method Using Kato's analytic perturbation theory for self-adjoint operators and embedding rectangular matrices into block self-adjoint operators.
result Closed-form expressions for the n-th order spectral variations of singular values. The paper studies invariant complex manifolds in holomorphic slow-fast systems.
problem Existence of invariant complex manifolds in holomorphic systems.
method Geometric singular perturbation theory, Fenichel and Briot-Bouquet theories.
result Conditions are provided to guarantee the existence of one-dimensional invariant complex manifolds.
Study on combustion theory solutions, proving nondegeneracy and stability in limit.
problem One-phase singular perturbation problem in combustion theory.
method Introduce density condition to preserve nondegeneracy, classify stable solutions.
result Global stable solutions have flat level sets in dimensions ≤ 4.
The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.
problem Preventing stable tangent cones for area-minimizing boundaries under specific metrics.
method Developed a perturbation theorem and used spectral theory and compactness arguments.
result A residual set of metrics on Sn+1 precludes linearly stable tangent cones for area-minimizing boundaries. 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.
New bifurcation found in perturbations of non-generic closed self-shrinkers.
problem Understanding the behavior of perturbations in non-generic closed self-shrinkers.
method Analyzing the mean curvature flow singularity transitions.
result Different types of singularity transitions based on perturbation direction.
The paper proves stability of certain singularities in integrable systems.
problem Stability of singularities in integrable systems under perturbations.
method Analytic and smooth perturbations of completely integrable systems, connectedness condition.
result Non-degenerate singular fibers are structurally stable under small perturbations.
Consider a supervised dataset D=[A∣b], where b is the outcome column, rows of D correspond to observations, and columns of A are the features of the dataset. A central problem in machine learning and pattern recognition is to select the most important features from D to be able to predic…
Study analyzes perturbations in singular subspaces under random noise.
problem Understanding singular vector and subspace changes in signal-plus-noise models.
method Generalized Davis-Kahan-Wedin theorem for any unitarily invariant norm, considering ℓ∞ and ℓ2,∞ bounds. result Fine-grained insights into singular vector and subspace perturbations, including ℓ∞ and ℓ2,∞ bounds. Study shows instability of naked singularities in perfect fluid models.
problem Instability of naked singularities in Einstein equations coupled with isothermal perfect fluid.
method Investigated spherically symmetric self-similar naked singularities under C1,α perturbations of an external massless scalar field. result Spherically symmetric self-similar naked singularities are unstable to trapped surface formation.
Paper proves families of singularities can be topologically trivialized.
problem Understanding behavior of singularities under small perturbations.
method Establishes sufficient conditions for embedded topological trivialization.
result New instances of topological stability, including μ-constant deformations. Spectral methods simplify data analysis, improving accuracy and stability.
problem Extracting meaningful information from noisy, incomplete data.
method Eigenvalues and eigenvectors of matrices constructed from data.
result Spectral methods are effective and can be analyzed using modern statistical theory.
Singular fiber resolution does not describe the spontaneous breaking of gauge symmetry in F-theory, as the corresponding branch of the moduli space does not exist in the theory. Accordingly, even non-abelian gauge theories have not been fully understood in global F-theory compactifications. We present a systematic disc…
We develop techniques for computing the integer valued SU(3) Casson invariant. Our method involves resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the perturbed flat moduli space. These techniques, together with Bott-Morse theory and the spl…
In dimension 7, we establish a Fredholm theory for a Dirac-type operator associated to a connection with point singularities. There are two applications. 1. over a closed 7-manifold, under some natural conditions, a G2−instanton and its point singularities can still be "seen" when the G2−structure is proper…
Novel mathematical approach using resurgent analysis reveals new structures in complex Chern-Simons theory.
problem Curious bijection in vertex algebras and SCFTs.
method Resurgent analysis, numerical algorithms, singularity elimination.
result New structures and patterns in complex Chern-Simons theory on hyperbolic 3-manifolds.
We use the twistorial construction of D-instantons in Calabi-Yau compactifications of type II string theory to compute an explicit expression for the metric on the hypermultiplet moduli space affected by these non-perturbative corrections. In this way we obtain an exact quaternion-Kahler metric which is a non-trivial d…
We extend the potential theory on almost minimzers from Part 1. We introduce so-called Hardy structures to study many classical operators using the tools from part 1. Furthermore, we show that for a naturally defined operator L, minimal growth of positive solutions of Lw = 0 towards the singular set is a stable propert…
We use closed geodesics to construct and compute Bott-type Morse homology groups for the energy functional on the loop space of flat n-dimensional tori, n≥1, and Bott-type Floer cohomology groups for their cotangent bundles equipped with the natural symplectic structure. Both objects are isomorpic to the singula…
New method proves instability of naked singularity and censors it.
problem Proving instability and censoring naked singularity.
method Einstein-scalar field system, hyperbolic short-pulse method, non-perturbative elliptic arguments.
result Tiny anisotropic perturbation leads to anisotropic apparent horizon censoring the naked singularity.
New technique stabilizes singular values in concatenated matrices.
problem How singular values of concatenated matrices relate to individual components.
method Developed perturbation technique extending classical results to concatenated matrices.
result Dominant singular values remain stable under small perturbations in submatrices.
The study shows stability of neckpinch singularities in mean curvature flows.
problem Stability of neckpinch singularities in mean curvature flows.
method Analysis of mean curvature flow and perturbations.
result Stability of neckpinch singularities in mean curvature flows.
Study 3d N=1 vacua from M-theory compactification on Spin(7) space.
problem Quantum corrections in 3d N=1 vacua from M-theory compactification.
method Use Higgs bundles to analyze 3d N=1 vacua and track corrections.
result Topological anomalies are robust and calculable in 3d effective field theory.
Perturbs area-minimizing hypersurfaces to reduce singular set's dimension.
problem Reduces the dimension of the singular set of area-minimizing hypersurfaces.
method Perturbs a smooth hypersurface to minimize the Minkowski dimension of the singular set.
result The singular set of the perturbed minimizing current has Minkowski dimension less than n-9.
Deterministic bounds for tensor singular values and vectors, differing from matrix cases.
problem Spectral learning of higher-order orthogonally decomposable tensors.
method Deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors.
result Perturbation affects each essential singular value/vector in isolation, independent of multiplicity and distance from other singular values.
Extract anomalies from 5D SCFTs using extra-dimensional η-invariants.
problem Anomalies in quantum field theories.
method Use extra-dimensional η-invariants to bypass traditional blowup techniques.
result Anomalies can be determined directly from η-invariants of asymptotic boundaries.
The Davis-Kahan-Wedin sinΘ theorem describes how the singular subspaces of a matrix change when subjected to a small perturbation. This classic result is sharp in the worst case scenario. In this paper, we prove a stochastic version of the Davis-Kahan-Wedin sinΘ theorem when the perturbation is a Gaussian rando…
Ricci flow simulations show unstable Fubini-Study metrics develop singularities.
problem Understanding the behavior of unstable perturbations in Ricci flow.
method Numerical simulations of Ricci flow starting from unstable Fubini-Study metrics.
result Ricci flow solutions from unstable Fubini-Study metrics develop local singularities.
Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.
problem Understanding neck pinches in Lagrangian flows.
method Introduced nondegenerate neck pinch and teardrop singularities, proving stability and answering questions.
result Nondegenerate neck pinches are stable and can be perturbed to nondegenerate singularities.
Numerical simulations show stability of Type-II singularities in noncompact hypersurfaces.
problem Stability of Type-II singularities in noncompact hypersurfaces with rotationally-symmetric perturbations.
method Adaptation of the overlap method to include angular dependence.
result MCF of noncompact hypersurfaces with angular dependence behaves similarly to rotationally-symmetric perturbations, developing Type-II or Type-I singularities.
Smooth bundles with rough data maintain Hodge kernel isomorphism.
problem Maintaining Hodge kernel isomorphism for smooth bundles with non-smooth geometric data.
method Analyzing nilpotent differential operators and Hodge-Dirac-type operators under perturbations of geometric data.
result Kernels of Hodge-Dirac operators remain isomorphic under uniform perturbations of geometric data.
The higher order singular value decomposition (HOSVD) of tensors is a generalization of matrix SVD. The perturbation analysis of HOSVD under random noise is more delicate than its matrix counterpart. Recently, polynomial time algorithms have been proposed where statistically optimal estimates of the singular subspaces …
Study shows instability of naked singularities in scalar field models.
problem Stability of naked singularities in spherically symmetric Einstein-Scalar field systems.
method Analysis of a family of incoming null cones becoming increasingly singular.
result Naked singularities are unstable to black hole formation under certain perturbations.
Singularities of area minimizing hypersurfaces can be smoothed in dimensions 9 and 10.
problem Singularities of area minimizing hypersurfaces.
method Perturbation of singularities.
result Singularities can be perturbed away in dimensions 9 and 10.
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.
Paper shows perturbed Taub-Bolt metric becomes singularity under Ricci flow.
problem Analyzing stability of Taub-Bolt metric under Ricci flow.
method Box argument and construction of Ricci flows on compact manifolds.
result Compact perturbation of Taub-Bolt metric evolves into finite time singularity.
The paper explores how polynomial roots and operator eigenvalues change with parameters.
problem How do roots of polynomials and eigenvalues of operators vary with parameter changes?
method Analyzes parameter dependence of polynomials and linear operators, covering real analytic to differentiable of finite order.
result Definitive optimal results for perturbation theory of polynomials and linear operators, including hyperbolic polynomials.
Proves conditions for complexification of real maps and their homology.
problem Conditions for homology and homotopy equivalence of real and complex maps.
method Analyzes local behavior of singular points and proves necessary conditions.
result Proves a conjecture about good real perturbations and their homotopy equivalence.
Study of mean curvature flow with obstacles using singular perturbation.
problem Obstacle problem associated to mean curvature flow.
method Geometric vanishing-viscosity approximation with singular perturbation.
result Generic level sets are distributional solutions of the obstacle problem.
Survey on Allen-Cahn equations and systems, focusing on multiplicity results and geometric interpretation.
problem Multiplicity results for Allen-Cahn equations and systems in singular perturbation regime.
method Photography method, variational-topological approach based on localized approximate solutions and barycenter maps.
result Encoding of topology into multiplicity results through variational-topological approach.
The paper explores hidden torus symmetries in integrable systems and their stability.
problem Structural stability of singularities in integrable systems.
method Use of hidden torus actions near singular orbits and integrable perturbations.
result Persistence of toric symmetries and structural stability of Kalashnikov's parabolic orbits.
The purpose of the paper is to introduce some conjectures regarding the analytic continuation and the arithmetic properties of quantum invariants of knotted objects. More precisely, we package the perturbative and nonperturbative invariants of knots and 3-manifolds into two power series of type P and NP, convergent in …
We construct the Seiberg-Witten theory on 3-manifolds with Euclidean ends (connected sums of R3 and a compact manifold) with perturbations which approximate ∗dx3 at infinity, and describe the structure of the moduli spaces. The setup is inspired by Taubes's program of relating the 4-dimensional Seiberg-Witten in…
We regard pre-trained residual networks (ResNets) as nonlinear systems and use linearization, a common method used in the qualitative analysis of nonlinear systems, to understand the behavior of the networks under small perturbations of the input images. We work with ResNet-56 and ResNet-110 trained on the CIFAR-10 dat…
We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degene…
The paper proves prevalent existence and partially determines moduli space of area-minimizing surfaces with fractal singular sets.
problem Existence and moduli space of area-minimizing surfaces with fractal singular sets.
method Proof of prevalent existence, determination of moduli space, refinement of strata.
result Sharp results on moduli space and refinement of strata, showing fractal singularities do not completely dissolve under generic perturbations.