This paper reverses a construction by merging boundary critical points into an interior one.
problem Pushing interior critical points to the boundary and splitting them into two boundary points.
method Specific assumptions allow merging two boundary critical points into one interior critical point.
result Merging two boundary critical points into a single interior critical point.
The minimal number of critical points is studied for smooth functions on closed manifolds.
problem Determining the minimal number of critical points for smooth functions on closed manifolds.
method Investigates cylindrical ball neighborhoods and exotic critical points, proving the conjecture for certain types of critical points.
result The minimal number of critical points is the same for smooth functions without exotic critical points on closed manifolds of dimension at least 6.
The study confirms a conjecture about critical points of smooth functions.
problem Understanding isolated critical points of smooth functions.
method Investigated cone-like, reasonable, and Rothe H hypothesis critical points.
result The conjecture holds true for certain critical points.
Hard to approximate critical points for simple nonconvex functions.
problem Approximating critical points of nonconvex functions.
method Proving hardness results for polynomial-time approximation of critical points.
result Proving that approximating critical points is intractable for simple nonconvex functions.
Symmetric critical points lead to symmetry breaking in neural networks.
problem Understanding symmetry in critical points of invariant functions.
method Analyzing the symmetry of critical points and their neighbors in invariant nonconvex functions.
result Symmetric critical points in invariant nonconvex functions are generically followed by symmetry breaking adjacent points.
Upper bounds found for systole function critical points on surface moduli space.
problem Finding upper bounds for systole function critical points.
method Analyzing the systole function on the surface moduli space.
result Upper bounds for critical points of systole function and their systole values.
Study critical points of Laplace eigenfunctions in polygons.
problem Characterize critical points of Laplace eigenfunctions in polygonal domains.
method Analyze components of the critical set with codimension 1.
result For simply connected polygons, if a second Neumann eigenfunction has infinitely many critical points, the polygon must be a rectangle.
The paper studies critical points in overparameterized neural networks, identifying a star locus and degenerate critical points.
problem Understanding the geometry of loss functions in overparameterized neural networks.
method Identifying and analyzing components of the critical locus of the loss function L for overparameterized feedforward neural networks of depth ℓ≥4. result For very wide networks, all critical points are degenerate, and lower bounds on the number of zero eigenvalues of the Hessian are given.
Noninjective monodromy found in polynomial critical point tracking.
problem Tracking critical points in polynomials leads to noninjective monodromy.
method Monic squarefree complex polynomials with prescribed critical point multiplicities.
result Monodromy map is noninjective for polynomials with exactly two critical points.
In this paper are studied the simplest patterns of axial curvature lines (along which the normal curvature vector is at a vertex of the ellipse of curvature) near a critical point of a surface mapped into R4. These critical points, where the rank of the mapping drops from 2 to 1, occur isolated in generic one parameter…
This paper focuses on the problem of topological equivalence of functions with isolated critical points on the boundary of a compact surface M which are also isolated critical points of their restrictions to the boundary. This class of functions we denote by Ω(M). Firstly, we've obtained the topological classificat…
The paper studies the smoothness of critical points of variational integrals on Hessian spaces.
problem The study focuses on the regularity of critical points of variational integrals defined on Hessian spaces.
method The approach involves solving a fourth order nonlinear equation and analyzing the Hessian of the critical points.
result Smooth critical points with bounded Hessian are shown to be smooth provided their Hessian has small BMO.
Study on critical points in random neural networks, revealing three regimes based on activation function.
problem Investigating the expected number of critical points in random neural networks.
method Deriving asymptotic formulas for critical points under infinite-width limit and suitable regularity conditions.
result Three distinct regimes of critical points behavior depending on activation function.
Numerically locating the critical points of non-convex surfaces is a long-standing problem central to many fields. Recently, the loss surfaces of deep neural networks have been explored to gain insight into outstanding questions in optimization, generalization, and network architecture design. However, the degree to wh…
The paper classifies and analyzes the stability of elastic curves with fixed endpoints.
problem Classification and stability of pinned elasticae.
method Critical points of the length-penalized elastic bending energy among planar curves with fixed endpoints.
result Explicit parametrization and classification of all critical points with a threshold parameter \(\hatλ \simeq 0.70107\).
Critical hypersurfaces with boundary have unique shapes and properties.
problem Characterizing the shapes of hypersurfaces with boundary and zero fractional mean curvature.
method Analyzing critical points of fractional area in RN with boundary conditions. result Critical hypersurfaces with specific boundary conditions are not simple shapes like (N−1)-balls. Study classifies ruled surfaces critical to Dirichlet energy.
problem Identifying ruled surfaces critical to Dirichlet energy.
method Explicit parametrization of ruled surfaces.
result Classification of ruled surfaces as critical points of Dirichlet energy.
We study critical points of holomorphic sections of $\ocal(m)$ on $\CP^n$. For quadrics, we give a complete discription of their critical points. When n=1, we prove a spherical Gauss-Lucas theorem. For general situation, we prove that a general section has all its critical points isolated and non-degenerate.
The exponential map fails to be injective near critical points in sub-Riemannian geometry.
problem Injectivity failure of the exponential map at critical points in sub-Riemannian geometry.
method Analysis of the Hilbert invariant integral of the variational problem associated with the sub-Riemannian structure.
result Characterization of conjugate points in terms of metric structure.
Study on critical faces convergence in a Poisson point process.
problem Convergence of point processes associated with critical faces in a Čech filtration.
method Established convergence in M0-topology for critical faces above vanishing threshold. result Obtained limit theorems for positive and negative critical faces.
Ricci solitons as critical points of quadratic curvature functionals
problem Einstein metrics and Ricci solitons as critical points of quadratic Riemannian functionals
method Study of Ricci solitons as critical points of a special quadratic curvature functional
result Ricci solitons are non-Einstein critical points of these functionals
Study classifies Morse functions with 4 critical points on immersed 2-spheres.
problem Classifying Morse functions with 4 critical points on immersed 2-spheres.
method Used dual graph of immersion and Reeb graphs to classify functions.
result Found all possible structures of the functions.
Paper proves convex domains have one maximum for semi-stable solutions.
problem Analyzing critical points of semi-stable solutions on convex domains.
method Relating critical points to an auxiliary function and using topological degree.
result Positive, semi-stable solutions have exactly one non-degenerate critical point.
Research characterizes critical points of scalar curvature functionals.
problem Characterizing critical points of scalar curvature functionals.
method Translation and analysis of a previous Russian paper.
result Provides insights into critical points of scalar curvature functionals.
To each isolated critical point of a smooth function on a 3-manifold we put in correspondence a tree (graph without cycles). We will prove that functions are topologically equivalent in the neighborhoods of critical points if and only if the corresponding trees are isomorphic. A complete topological invariant of functi…
The paper proves conditions under which critical point metrics are Einstein.
problem Proving that critical point metrics are Einstein under specific curvature constraints.
method Analyzing the traceless Ricci operator and scalar curvature constraints.
result The conjecture that critical point metrics are Einstein is proven under certain curvature conditions.
Investigates energy minimizers and critical points of scale-invariant tangent-point energies for knots.
problem Finding and characterizing minimizers and critical points of scale-invariant tangent-point energies for closed curves.
method Develops convergence and regularity theories based on fractional Sobolev spaces and new energy functionals.
result Minimizing sequences converge to locally critical embeddings in all but finitely many points, and locally critical embeddings are regular.
The paper simplifies complex 2D functions near their critical points.
problem Simplifying smooth functions on 2-manifolds near critical points.
method Explicit construction of coordinate changes to canonical form.
result Estimates the radius of required neighbourhoods for specific singularity types.
Stability of a new map derived from the equator map is analyzed.
problem Stability of a new map derived from the equator map.
method Detailed stability analysis of the generalized equator map as a critical point of the extrinsic k-energy and p-energy.
result Established generalizations of classical (in)stability results.
Riemannian gradient descent escapes some spurious critical points on low-rank matrix manifold.
problem Spurious critical points on the boundary of low-rank matrix manifold.
method Riemannian gradient descent with dynamical low-rank approximation and rescaled gradient flow.
result Riemannian gradient descent escapes some spurious critical points on the boundary of the manifold.
The main result of this paper is a construction of solutions to the reverse Yang-Mills-Higgs flow converging in the C∞ topology to a critical point. The construction uses only the complex gauge group action, which leads to an algebraic classification of the isomorphism classes of points in the unstable set of a…
Near a birth-death critical point in a one-parameter family of gradient flows, there are precisely two Morse critical points of index difference one on the birth side. This paper gives a self-contained proof of the folklore theorem that these two critical points are joined by a unique gradient trajectory up to time-shi…
Study on smoothness of 4D Willmore-type hypersurfaces.
problem Investigating smoothness of critical points of a 4D Willmore-type energy.
method Computed first variation, applied Noether's theorem, investigated other generalizations.
result Critical points of the energy are smooth.
New result on critical points of Bethe free energy under deformation retracts.
problem Characterizing critical points of Bethe free energy for complex graphs.
method Analyzing homotopy types and deformation retracts of factor graphs.
result Critical points of Bethe free energy are invariant under deformation retracts.
Investigates a new four-dimensional energy related to Willmore energy.
problem Exploring a new conformally invariant energy in four dimensions.
method Computed first variation, applied Noether's theorem, investigated other generalizations.
result Critical points of the new energy are smooth and do not include minimal hypersurfaces.
We consider functions with isolated critical points on a closed surface. We prove that in a neighborhood of a critical point the function conjugates with Rezk for the some nonnegative integer k. The full topological invariant of such functions is constructed.
Examples of smooth maps with finitely many critical points in dimensions (4,3), (8,5) and (16,9)math.GT We consider manifolds M2n which admit smooth maps into a connected sum of S1×Sn with only finitely many critical points, for n∈{2,4,8}, and compute the minimal number of critical points.
The paper proves that Gaussian field critical points have finite moments.
problem Proving the finiteness of moments for Gaussian field critical points.
method General approach not specific to critical points, using Taylor polynomial non-degeneracy.
result The finiteness of moments of the number of critical points of Gaussian fields.
Critical points of scale-invariant curvature energies in 4D are analytic.
problem Analyzing critical points of curvature energies in 4D manifolds.
method Applying Noether's theorem to identify conservation laws and lower order elliptic system of PDEs, then using integrability by compensation and interpolation theory.
result Critical points of scale-invariant curvature energies in 4D are analytic.
Jakobson and Nadirashvili \cite{JN} constructed a sequence of eigenfunctions on T2 with a bounded number of critical points, answering in the negative the question raised by Yau \cite{Yau1} which asks that whether the number of the critical points of eigenfunctions for the Laplacian increases with the corresponding …
Characterizes critical points in convex double and triple bubbles.
problem Critical points of double and triple bubbles in convex shapes.
method Characterization through stationary varifolds in Rn and R3. result Characterization of critical points in convex shapes.
Study stability of non-Kähler Calabi-Yau metrics using critical points of generalized Einstein Hilbert action.
problem Stability of critical points of the generalized Einstein Hilbert action in non-Kähler Calabi-Yau theory.
method Analysis of Bismut Hermitian Einstein manifolds and Bismut flat pluriclosed steady solitons, proving stability conditions.
result All Bismut Hermitian Einstein manifolds are linearly stable, and all Bismut flat pluriclosed steady solitons with positive Ricci curvature are linearly strictly stable.
Bound critical points for minimal Radó functions.
problem Counting interior critical points for minimal Radó functions.
method Bounding critical points in terms of boundary data and domain Euler characteristic.
result Bound the number of interior critical points.
Study uncovers complex critical points in tensor decomposition.
problem Nonconvex optimization of symmetric tensor decomposition.
method Utilized symmetry to construct critical points and analyze Hessian.
result Obtained precise analytic estimates on objective function and Hessian.
Paper proves conjecture about Einstein metrics on manifolds with positive isotropic curvature.
problem Proving the Besse conjecture for metrics with positive isotropic curvature.
method Analyzing the critical point equation and using properties of metrics with positive isotropic curvature.
result The Besse conjecture is true for metrics with positive isotropic curvature.
Classical Ljusternik-Schnirelmann category is upper bounded by the number of critical points of any bounded from below differentiable functions of Palais-Smale type. Here we achieve an adaptation of this result for the tangential category of foliations. We introduce a weaker type of Palais-Smale function, obtaining a s…
Study finds critical points in perimeter functional for fixed volume sets.
problem Finding critical points in perimeter functional for sets of fixed volume.
method Utilizes Mazurwoski--Zhou techniques and new Cacciopoli set connectedness results.
result Constructs smooth almost embedded hypersurfaces with non-zero constant mean curvature.
Paper proves Łojasiewicz inequalities near simple bubble trees on surfaces.
problem Proving Łojasiewicz inequalities for critical points on surfaces.
method Deriving sufficient conditions for Łojasiewicz inequalities near almost-critical points in a Hilbert space.
result Sequences of almost critical points satisfy Łojasiewicz inequalities as they approach the first non-trivial bubble tree.