Sublevel Sets Research Articles

Kinematic formulas for sets defined by differences of convex functions

The class WDC(M) consists of all subsets of a smooth manifold M that may be expressed in local coordinates as sufficiently regular sublevel sets of DC (differences of convex) functions. If M is Riemannian and G is a group of isometries acting transitively on the sphere bundle SM, we define the invariant curvature measures of compact WDC subsets of M, and show that pairs of such subsets are subject to the array of kinematic formulas known to apply to smoother sets. Restricting to the case (M,G)=(Rd,SO(d)‾), this extends and subsumes Federer's theory of sets with positive reach in an essential way. The key technical point is equivalent to a sharpening of a classical theorem of Ewald, Larman, and Rogers characterizing the dimension of the set of directions of line segments lying in the boundary of a given convex body.

Gerstewitz Functionals on Linear Spaces and Functionals with Uniform Sublevel Sets

In this paper, we study Gerstewitz functionals that are defined on an arbitrary linear space without assuming any topology. Extended real-valued functions with uniform sublevel sets turn out to be Gerstewitz functionals if the sublevel sets can be described by a linear shift of a set in a specified direction. Gerstewitz functionals can represent binary relations and thus act as a tool for scalarization. Sets, which are not necessarily convex, can be separated by Gerstewitz functionals. Conditions are given under which a Gerstewitz functional is finite-valued, convex, positively homogeneous, subadditive, sublinear or monotone. The values of each Gerstewitz functional are connected with those of a sublinear function. It is shown that some Minkowski functionals--especially order unit norms--coincide with a Gerstewitz functional on a subset of the space.

Lower semicontinuous functionals with uniform sublevel sets

In this paper, lower semicontinuous functionals with uniform sublevel sets are investigated, where the sublevel sets are linear shifts of a set in a fixed direction. The extended real-valued functionals are defined on a topological vector space. Conditions are given under which they are proper, finite-valued, continuous, convex, sublinear, strictly quasi-convex, strictly quasi-concave or monotone. We apply the functionals to the separation of not necessarily convex sets.

Robust topological inference

Let P be a distribution with support S. The salient features of S can be quantified with persistent homology, which summarizes topological features of the sublevel sets of the distance function (th...

Topological Network Analysis of Electroencephalographic Power Maps.

Meditation practice as a non-pharmacological intervention to provide health related benefits has generated much neuroscientific interest in its effects on brain activity. Electroencephalogram (EEG), an imaging modality known for its inexpensive procedure and excellent temporal resolution, is often utilized to investigate the neuroplastic effects of meditation under various experimental conditions. In these studies, EEG signals are routinely mapped on a topographic layout of channels to visualize variations in spectral powers within certain frequency ranges. Topological data analysis (TDA) of the topographic power maps modeled as graphs can provide different insight to EEG signals than standard statistical methods. A highly effective TDA technique is persistent homology, which reveals topological characteristics of a power map by tracking feature changes throughout a filtration process on the graph structure of the map. In this paper, we propose a novel inference procedure based on filtrations induced by sublevel sets of the power maps of high-density EEG signals. We apply the pipeline to simulated and real data, where we compare the persistent homological features of topographic maps of spectral powers in high-frequency bands of EEG signals recorded on long-term meditators and meditation-naive practitioners.

Verification estimates for the construction of Lyapunov functions using meshfree collocation

Lyapunov functions are functions with negative derivative along solutions of a given ordinary differential equation. Moreover, sub-level sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. One of the numerical construction methods for Lyapunov functions uses meshfree collocation with radial basis functions (RBF). In this paper, we propose two verification estimates combined with this RBF construction method to ensure that the constructed function is a Lyapunov function. We show that this combination of the RBF construction method and the verification estimates always succeeds in constructing and verifying a Lyapunov function for nonlinear ODEs in $ \mathbb{R}^d $ with an exponentially stable equilibrium.

A new multicomponent Poincaré–Beckner inequality

We prove a new vectorial functional inequality of Poincaré–Beckner type. The inequality may be interpreted as an entropy–entropy production one for a gradient flow in the metric space of Radon measures. The proof uses subtle analysis of combinations of related super- and sub-level sets employing the coarea formula and the relative isoperimetric inequality.

The Morse theory of Čech and Delaunay complexes

Given a finite set of points in $\mathbb R^n$ and a radius parameter, we study the \v{C}ech, Delaunay-\v{C}ech, Delaunay (or Alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the \v{C}ech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, which are explicitly described by a single discrete gradient field.

Global solvability of real analytic involutive systems on compact manifolds

The focus of this work is the smooth global solvability of a linear partial differential operator $${\mathbb {L}}$$ associated to a real analytic closed non-exact 1-form b—defined on a real analytic closed n-manifold—that may be naturally regarded as the first operator of the complex induced by a locally integrable structure of tube type and co-rank one. We define an appropriate covering projection $$\widetilde{M}\rightarrow M$$ such that the pullback of b has a primitive $$\widetilde{B}$$ and prove that the operator is globally solvable if and only if the superlevel and sublevel sets of $$\widetilde{B}$$ are connected. As a byproduct we obtain a new geometric characterization for the global hypoellipticity of the operator. When M is orientable we prove a connection between the global solvability of $${\mathbb {L}}$$ and that of $${\mathbb {L}}^{n-1}$$ which is the last non-trivial operator of the complex, in particular, we prove that $${\mathbb {L}}$$ is globally solvable if and only if $${\mathbb {L}}^{n-1}$$ is globally solvable. In the smooth category, we are able to provide analogous geometric characterizations of the global solvability and the global hypoellipticity when b is a Morse 1-form, i.e., when the structure is of Mizohata type.

Maximal decay inequalities for trilinear oscillatory integrals of convolution type

In this paper we prove sharp L∞–L∞–L∞ decay for certain trilinear oscillatory integral forms of convolution type on R2. These estimates imply earlier L2–L2–L2 results obtained by the second author as well as corresponding sharp, stable sublevel set estimates of the form studied by Christ [3] and Christ, Li, Tao, and Thiele [5]. New connections to the multilinear results of Phong, Stein, and Sturm [13] are also considered.

Nonlinear Error Bounds via a Change of Function

This work can be seen as a sequel to our previous paper, which dealt with local nonlinear error bounds for lower semicontinuous functions on complete metric spaces, based on estimates of the strong slope of the function through the distance to a sublevel set. Here, we consider estimates of the strong slope through the values of the function, and we provide characterizations of nonlinear local and global error bounds that are again obtained by a reduction to the linear case. Our main tool is a simple chain rule for the strong slope. We provide several examples showing that recent results in the literature can be recaptured from our general framework.

Choquet-Monge-Ampère Classes

We introduce and study Choquet-Monge-Ampere classes on compact Kahler manifolds. They consist of quasi-plurisubharmonic functions whose sublevel sets have small enough asymptotic Monge-Ampere capacity. We compare them with finite energy classes, which have recently played an important role in Kahler Geometry.

On Some Basic Results Related to Affine Functions on Riemannian Manifolds

We study some basic properties related to affine functions on Riemannian manifolds. A characterization for a function to be linear affine is given and a counterexample on Poincare plane is provided, which, in particular, shows that assertions (i) and (ii) claimed by Papa Quiroz in (J Convex Anal 16(1):49---69, 2009, Proposition 3.4) are not true, and that the function involved in assertion (ii) is indeed not quasi-convex. Furthermore, we discuss the convexity properties of the sub-level sets of the function on Riemannian manifolds with constant sectional curvatures.

Approximation of reachable sets using optimal control and support vector machines

We propose and discuss a new computational method for the numerical approximation of reachable sets for nonlinear control systems. It is based on the support vector machine algorithm and represents the set approximation as a sublevel set of a function chosen in a reproducing kernel Hilbert space. In some sense, the method can be considered as an extension to the optimal control algorithm approach recently developed by Baier, Gerdts and Xausa. The convergence of the method is illustrated numerically for selected examples.

On the global solvability of involutive systems

We consider a class of involutive systems in Tn+1 associated with a closed 1-form defined on the torus Tn. We prove that, under a geometric condition, the global solvability of this class is equivalent to a diophantine condition involving Liouville forms and the connectedness of all sublevel and superlevel sets of a global primitive associated with the system.

Adaptive subgradient method for the split quasi-convex feasibility problems

In this paper, we consider a type of the celebrated convex feasibility problem, named as split quasi-convex feasibility problem (SQFP). The SQFP is to find a point in a sublevel set of a quasi-convex function in one space and its image under a bounded linear operator is contained in a sublevel set of another quasi-convex function in the image space. We propose a new adaptive subgradient algorithm for solving SQFP problem. We also discuss the convergence analyses for two cases: the first case where the functions are upper semicontinuous in the setting of finite dimensional, and the second case where the functions are weakly continuous in the infinite-dimensional settings. Finally some numerical examples in order to support the convergence results are given.

Distributed partitioning algorithms for multi-agent networks with quadratic proximity metrics and sensing constraints

We consider a class of Voronoi-like partitioning problems, in which a multi-agent network seeks to subdivide a subset of an affine space into a finite number of cells in the presence of sensing constraints. The cell of this subdivision that is assigned to a particular agent consists exclusively of points that can be sensed by this agent and are closer to it than to any other agent that can also sense them. The proximity between an agent and an arbitrary point is measured in terms of a non-homogeneous quadratic (generalized) distance function, which does not, in general, enjoy the triangle inequality and the symmetry property. One of the consequences of this fact is that the structure of the sublevel sets of the utilized proximity metric does not conform with that of the sensing region of an agent. Due to this mismatch, it is possible that a point may be assigned to an agent which is different from its “nearest” agent simply because the nearest agent cannot sense this point, unless special care is taken. We propose a distributed partitioning algorithm that enables each agent to compute its own cell independently from the other agents when the only information available to it is the positions and the velocities of the agents that lie inside its sensing region. The algorithm is based on an iterative process that adjusts the size of the sensing region of each agent until the associated cell of the latter corresponds to the intersection of its sensing region with the cell that would have been assigned to it in the absence of sensing constraints. The correctness of the proposed distributed algorithm, which successfully handles the aforementioned issues, is studied in detail.

Existence and Regularity of Periodic Solutions to Certain First-Order Partial Differential Equations

We present conditions on the coefficients of a class of vector fields on the torus which yield a characterization of global solvability as well as global hypoellipticity, in other words, the existence and regularity of periodic solutions. Diophantine conditions and connectedness of certain sublevel sets appear in a natural way in our results.

Sufficient conditions for error bounds of difference functions and applications

This paper establishes verifiable sufficient conditions for the existence of error bounds for the sub-level set of a difference function over an abstract constraint by applying a technique used by A. D. Ioffe. As a consequence, error bounds for constraint systems defined by d.c. inequalities and their applications in studying of exactness of the associated $$\ell _1$$l1 penalty function and existence of Lagrange multipliers as necessary optimality conditions are also investigated.

Local stability and local stabilization of discrete-time T–S fuzzy systems with time-delay

The goal of this paper is to propose linear matrix inequality (LMI) formulations to analyze the local stability and local stabilization of discrete-time nonlinear systems represented by Takagi–Sugeno (T–S) fuzzy systems with time-delay. In order to estimate the domain of attraction (DA) of the time-delay systems, the so-called fuzzy Lyapunov-Krasovskii functionals are used to characterize the subsets of the DA as sublevel sets of the Lyapunov- Krasovskii functionals. We use the information of the variation rates of the membership functions to obtain less conservative results. Finally, examples are given to illustrate the proposed methods.