Tangent Cone Research Articles

Cohomology of contact loci

We construct a spectral sequence converging to the cohomology with compact support of the m-th contact locus of a complex polynomial. The first page is explicitly described in terms of a log resolution and coincides with the first page of McLean’s spectral sequence converging to the Floer cohomology of the $m$‑th iterate of the monodromy, when the polynomial has an isolated singularity. Inspired by this connection, we conjecture that if two germs of holomorphic functions are embedded topologically equivalent, then the Milnor fibers of their tangent cones are homotopy equivalent.

Gluing of multiple Alexandrov spaces

Perel'man's Doubling Theorem [11] says that the doubling of an Alexandrov space is also an Alexandrov space with the same lower curvature bound. Petrunin [12] generalizes this theorem to two Alexandrov spaces being glued along their isometric boundaries. These theorems have a lot of applications to the study of spaces with lower curvature bounds. As a generalization to these results and a tool to construct and verify new Alexandrov spaces, we study the gluing of multiple Alexandrov spaces. We consider a Gluing Conjecture, which says that the gluing of finite number of Alexandrov spaces is an Alexandrov space, if and only if the gluing is by path isometry along the boundaries and the tangent cones are glued to be Alexandrov spaces.We prove the Gluing Conjecture when the complement of (n−1,ϵ)-regular points is discrete in the glued part. In particular, this implies the Gluing Conjecture in dimension 2. This is a new result in the case of gluing 2-dimensional convex sets. In our proof, we also establish some structural theory for the general gluing. Based on the tangent cone condition, the 2-point gluing over (n−1,ϵ)-regular points is classified to be locally separable, as being assumed in Petrunin's Gluing Theorem. The gluing near an un-glued (n−1,ϵ)-regular point is classified to be involutional.

Calabi‐Yau metrics with cone singularities along intersecting complex lines: The unstable case

We produce local Calabi-Yau metrics on $\mathbf C^2$ with conical singularities along three or more complex lines through the origin whose cone angles strictly violate the Troyanov condition. The tangent cone at the origin is a flat polyhedral K\"ahler cone with conical singularities along two intersecting lines: one with cone angle corresponding to the line with smallest cone angle, while the other forms as the collision of the remaining lines into a single conical line. Using a branched covering argument, we can construct Calabi-Yau metrics with cone singularities along cuspidal curves with cone angle in the unstable range.

Correction to: On the Continuity of the Tangent Cone to the Determinantal Variety

Correction to: On the Continuity of the Tangent Cone to the Determinantal Variety

Second-Order Optimality Conditions for Nonconvex Set-Constrained Optimization Problems

In this paper, we study second-order optimality conditions for nonconvex set-constrained optimization problems. For a convex set-constrained optimization problem, it is well known that second-order optimality conditions involve the support function of the second-order tangent set. In this paper, we propose two approaches for establishing second-order optimality conditions for the nonconvex case. In the first approach, we extend the concept of the support function so that it is applicable to general nonconvex set-constrained problems, whereas in the second approach, we introduce the notion of the directional regular tangent cone and apply classical results of convex duality theory. Besides the second-order optimality conditions, the novelty of our approach lies in the systematic introduction and use, respectively, of directional versions of well-known concepts from variational analysis.

On the Continuity of the Tangent Cone to the Determinantal Variety

Tangent and normal cones play an important role in constrained optimization to describe admissible search directions and, in particular, to formulate optimality conditions. They notably appear in various recent algorithms for both smooth and nonsmooth low-rank optimization where the feasible set is the set $\mathbb {R}_{\le r}^{m \times n}$ of all m × n real matrices of rank at most r. In this paper, motivated by the convergence analysis of such algorithms, we study, by computing inner and outer limits, the continuity of the correspondence that maps each $X \in \mathbb {R}_{\le r}^{m \times n}$ to the tangent cone to $\mathbb {R}_{\le r}^{m \times n}$ at X. We also deduce results about the continuity of the corresponding normal cone correspondence. Finally, we show that our results include as a particular case the a-regularity of the Whitney stratification of $\mathbb {R}_{\le r}^{m \times n}$ following from the fact that this set is a real algebraic variety, called the real determinantal variety.

Lipschitz Normally Embedded Set and Tangent Cones at Infinity

We prove that any analytic set in $\C^n$ with a unique tangent cone at infinity is an algebraic set. We prove that the degree of a complex algebraic set in $\C^n$, which is Lipschitz normally embedded at infinity, is equal to the degree of its tangent cone at infinity.

Stable Recovery and Separation of Structured Signals by Convex Geometry

We consider the ill-posed problem of identifying the signal that is structured in some general dictionary (i.e., possibly redundant or over-complete matrix) from corrupted measurements, where the corruption is structured in another general dictionary. We formulate the problem by applying appropriate convex constraints to the signal and corruption according to their structures and provide conditions for exact recovery from structured corruption and stable recovery from structured corruption with added stochastic bounded noise. In addition, this paper provides estimates of the number of the measurements needed for recovery. These estimates are based on computing the Gaussian complexity of a tangent cone and the Gaussian distance to a subdifferential. Applications covered by the proposed programs include the recovery of signals that is disturbed by impulse noise, missing entries, sparse outliers, random bounded noise, and signal separation. Numerical simulation results are presented to verify and complement the theoretical results.

On the tangential cone condition for electrical impedance tomography

We state some sufficient criteria for the tangential cone conditions to hold for the electrical impedance tomography problem. The results are based on Löwner convexity of the forward operator. As a consequence, we show that for conductivities that satisfy various properties, such as Hölder source conditions, finite-dimensionality, or certain monotonicity criteria, the tangential cone condition is verified.

Carnot metrics, dynamics and local rigidity

AbstractThis paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot–Carathéodory metric. Principally, we employ the theory of Margulis and Mostow, Métivier, Mitchell, and Pansu on tangent cones to establish resonances between Lyapunov exponents. We apply these results in three different settings. First, we explore rigidity properties of smooth dominated splittings for Anosov diffeomorphisms and flows via associated smooth Carnot–Carathéodory metrics. Second, we obtain local rigidity properties of higher hyperbolic rank metrics in a neighborhood of a locally symmetric one. For the latter application we also prove structural stability of the Brin–Pesin asymptotic holonomy group for frame flows. Finally, we obtain local rigidity properties for uniform lattice actions on the ideal boundary of quaternionic and octonionic symmetric spaces.

Canonical Identification at Infinity for Ricci-Flat Manifolds

We give a natural way to identify between two scales, potentially arbitrarily far apart, in a non-compact Ricci-flat manifold with Euclidean volume growth when a tangent cone at infinity has smooth cross section. The identification map is given as the gradient flow of a solution to an elliptic equation.

Levenberg–Marquardt method for ill-posed inverse problems with possibly non-smooth forward mappings between Banach spaces

In this paper, we consider a Levenberg–Marquardt method with general regularization terms that are uniformly convex on bounded sets to solve the ill-posed inverse problems in Banach spaces, where the forward mapping might not Gâteaux differentiable and the image space is unnecessarily reflexive. The method therefore extends the one proposed by Jin and Yang in (2016 Numer. Math. 133 655–684) for smooth inverse problem setting with globally uniformly convex regularization terms. We prove a novel convergence analysis of the proposed method under some standing assumptions, in particular, the generalized tangential cone condition and a compactness assumption. All these assumptions are fulfilled when investigating the identification of the heat source for semilinear elliptic boundary-value problems with a Robin boundary condition, a heat source acting on the boundary, and a possibly non-smooth nonlinearity. Therein, the Clarke subdifferential of the non-smooth nonlinearity is employed to construct the family of bounded operators that is a replacement for the non-existing Gâteaux derivative of the forward mapping. The efficiency of the proposed method is illustrated with a numerical example.

On the Betti numbers of the tangent cones for Gorenstein monomial curves

The aim of the article is to study the Betti numbers of the tangent cone of Gorenstein monomial curves in affine 4-space. If $C_S$ is a noncomplete intersection Gorenstein monomial curve whose tangent cone is Cohen--Macaulay, we show that the possible Betti sequences are (1,5,5,1), (1,5,6,2) and (1,6,8,3).

Levenberg–Marquardt method for solving inverse problem of MRE based on the modified stationary Stokes system

Magnetic resonance elastography (MRE) is a developing medical imaging technique which can successfully diagnose liver diseases such as cirrhotic and fibrosis. As a hardware MRE gives an image of a viscoelastic wave in a region of interest (ROI) of a human body. Here the wave is generated by an external vibration system and further transmitted to the human body through a probe. MRE also has a software usually called elastogram which reconstructs viscoelastic modulus from the wave displacement vector. An appropriate model equation or system which connects the wave image to the modulus is known as modified stationary Stokes system if viscoelastic property of the ROI is isotropic (Ammari et al 2008 Q. Appl. Math. 66 139–175; Jiang et al 2011 SIAM J. Appl. Math. 71 1965–1989). Comparing this system with the original stationary Stokes system, it has an extra term with frequency. In most cases the elastogram has been using a more simpler model equation such as scalar model (Manduca et al 2003 Med. Image Anal. 5 237–254). In this paper we discuss about a scheme of solving the equation (*)F(A) = u, where u is the measured wave displacement and F(A) is the solution to the boundary value problem for the modified stationary Stokes system with the modulus A (see subsection for more details). More precisely, we will show the convergence of a Newton type iteration scheme called the Levenberg–Marquardt method by proving that the nonlinear operator F satisfies the tangential cone condition. We will also provide several numerical tests even with noisy data.

Separation characteristics and structure optimization of double spherical tangent double-field coupling demulsifier

The demulsification and dewatering of the W/O emulsion are widely used in petrochemical industry, oilfield exploitation, and resource and environmental engineering. However, efficiently treating emulsion via traditional single methods. In this study, a new double-field coupling demulsification and dewatering device is proposed, where the conical structure of the device is double spherical tangential type. The numerical model for double-field coupling is established, especially, the population balance model (PBM) is used to simulate the coalescence and breakup of dispersed droplets under the double-field coupling action. And the effects of three conical structures on the internal flow and separation efficiency are analyzed. Results show that the conical structure has a significant effect on the coalescence of droplets, especially the double spherical tangential cone is more conducive for improving the coalescence ability of small droplets and improving the separation efficiency of the device. After optimization, the optimal R value of the double spherical tangential coupling device is 300 mm, and the separation efficiency can be up to 96.32%, which is 6.13% higher than the separation efficiency of the straight double-cone coupling device.

Determining a time-dependent coefficient in a time-fractional diffusion-wave equation with the Caputo derivative by an additional integral condition

This paper is devoted to recovering a time-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation with the time Caputo derivative from an additional integral condition. The uniqueness and a conditional stability for such an inverse problem are proved. Then the two-point gradient method is used to solve the inverse zeroth-order coefficient problem numerically. Some properties of the forward operator are obtained, such as the Fréchet differentiability, the Lipschitz continuity and the tangential cone condition to guarantee the convergence of the proposed algorithm. Four numerical examples in one-dimensional and two-dimensional spaces are provided to show the effectiveness and stability of the suggested algorithm.

Minimizing Curves in Prox-regular Subsets of Riemannian Manifolds

We obtain a characterization of the proximal normal cone to a prox-regular subset of a Riemannian manifold and some properties of Bouligand tangent cones to these sets are presented. Moreover, we show that on an open neighborhood of a prox-regular set, the metric projection is locally Lipschitz and it is directionally differentiable at the boundary points of the set. Finally, a necessary condition for a curve to be a minimizing curve in a prox-regular set is derived.

Angle Sums of Schläfli Orthoschemes

We consider the simplices KnA={x∈Rn+1:x1≥x2≥⋯≥xn+1,x1-xn+1≤1,x1+⋯+xn+1=0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} K_n^A=\\{x\\in {\\mathbb {R}}^{n+1}:x_1\\ge x_2\\ge \\cdots \\ge x_{n+1},x_1-x_{n+1}\\le 1,\\,x_1+\\cdots +x_{n+1}=0\\} \\end{aligned}$$\\end{document}and KnB={x∈Rn:1≥x1≥x2≥⋯≥xn≥0},\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} K_n^B=\\{x\\in {\\mathbb {R}}^n:1\\ge x_1\\ge x_2\\ge \\cdots \\ge x_n\\ge 0\\}, \\end{aligned}$$\\end{document}which are called the Schläfli orthoschemes of types A and B, respectively. We describe the tangent cones at their j-faces and compute explicitly the sums of the conic intrinsic volumes of these tangent cones at all j-faces of K_n^A and K_n^B. This setting contains sums of external and internal angles of K_n^A and K_n^B as special cases. The sums are evaluated in terms of Stirling numbers of both kinds. We generalize these results to finite products of Schläfli orthoschemes of type A and B and, as a probabilistic consequence, derive formulas for the expected number of j-faces of the Minkowski sums of the convex hulls of a finite number of Gaussian random walks and random bridges. Furthermore, we evaluate the analogous angle sums for the tangent cones of Weyl chambers of types A and B and finite products thereof.

Comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces

We give an analogue of triangle comparison for Kähler manifolds with a lower bound on the holomorphic bisectional curvature. We show that the condition passes to noncollapsed Gromov–Hausdorff limits. We discuss tangent cones and singular Kähler spaces.

On the geometry of asymptotically flat manifolds

In this paper, we investigate the geometry of asymptotically flat manifolds with controlled holonomy. We show that any end of such manifold admits a torus fibration over an ALE end. In addition, we prove a Hitchin-Thorpe inequality for oriented Ricci-flat $4$-manifolds with curvature decay and controlled holonomy. As an application, we show that any complete asymptotically flat Ricci-flat metric on a $4$-manifold which is homeomorphic to $\mathbb R^4$ must be isometric to the Euclidean or the Taub-NUT metric, provided that the tangent cone at infinity is not $\mathbb R \times \mathbb R_+$.