Sublevel Sets Research Articles

Lelong-Demailly Numbers in Terms of Capacity and Weak Convergence for Closed Positive Currents

In this paper we give a new definition of the Lelong-Demailly number in terms of the CT-capacity, where T is a closed positive current and CT is the capacity associated to T. This derived from some esimate on the growth of the CT-capacity of the sublevel sets of a weighted plurisubharmonic (psh for short) function. These estimates enable us to give another proof of the Demailly's comparaison theorem as well as a generalization of some results due to Xing concerning the characterization of bounded psh functions. Another problem that we consider here is related to the existence of a psh function v that satisfies the equality CT (K) = ∫K T ∧ (ddcv)p, where K is a compact subset. Finally, we give some conditions on the capacity CT that guarantees the weak convergence ukTk → uT, for positive closed currents T, Tk and psh functions uk u.

Clarke Subdifferentials of a Minimal Time Function in Normed Spaces

In a general normed vector space, we study the minimal time function determined by a differential inclusion where the set-valued mapping involved has constant values of a bounded closed convex set U and by a closed target set S. We show that Clarke subdifferential of aminimal time function are representable by virtue of corresponding normal cones of sublevel set of the function and level or suplevel sets of the support function of U. The known results in the literature was obtained only in Banach space.

A multi-dimensional resolution of singularities with applications to analysis

We formulate a resolution of singularities algorithm for analyzing the zero sets of real-analytic functions in dimensions $\geq 3$. Rather than using the celebrated result of Hironaka, the algorithm is modeled on a more explicit and elementary approach used in the contemporary algebraic geometry literature. As an application, we define a new notion of the height of real-analytic functions, compute the critical integrability index, and obtain sharp growth rate of sublevel sets. This also leads to a characterization of the oscillation index of scalar oscillatory integrals with real-analytic phases in all dimensions.

Recovering an Homogeneous Polynomial from Moments of Its Level Set

Let $$\mathbf{K }:=\left\{ \mathbf{x }: g(\mathbf{x })\le 1\right\} $$K:=x:g(x)≤1 be the compact (and not necessarily convex) sub-level set of some homogeneous polynomial $$g$$g. Assume that the only knowledge about $$\mathbf{K }$$K is the degree of $$g$$g as well as the moments of the Lebesgue measure on $$\mathbf{K }$$K up to order $$2d$$2d. Then the vector of coefficients of $$g$$g is the solution of a simple linear system whose associated matrix is nonsingular. In other words, the moments up to order $$2d$$2d of the Lebesgue measure on $$\mathbf{K }$$K encode all information on the homogeneous polynomial $$g$$g that defines $$\mathbf{K }$$K (in fact, only moments of order $$d$$d and $$2d$$2d are needed).

Extended radial epiderivatives of non-convex vector-valued maps and parametric quasiconvex programming

In this paper, we consider extended vector-valued mappings defined on a normed linear space. Based on the recent semicontinuous regularizations related to hypographical and/or epigraphical profile mappings of the considered function introduced, we define semicontinuous radial epiderivatives. We, then, demonstrate that the properties of these epiderivatives amount to properties of hypographical and/or epigraphical profile mappings of the corresponding difference quotient of the underlying function, which simplify fairly well the proofs in the radial epiderivative formulaes. In particular, we stress the impact of semicontinuity, hence, we characterize with new arguments the radial epiderivatives in terms of the suprema and/or infima of the interiorly radial cone of the hypograph and/or epigraph of the considered function. Finally, we obtain optimality conditions for general non-convex constrained vector optimization problems. We apply thereafter the obtained pattern to a parametric quasiconvex programming problem for which we derive necessary and sufficient optimality conditions that are not sensitive to perturbation at the nominal level, yielding henceforth more – and strong at least under asymptotically regular constraints – information than the recent stability results obtained under additional conditions on the regularity of the normal cone to the adjusted sublevel sets of the underlying function.

Local stability analysis of continuous-time Takagi–Sugeno fuzzy systems: A fuzzy Lyapunov function approach

This paper proposes a strategy to estimate invariant subsets of the domain of attraction (DA) for asymptotically stable zero equilibrium points of continuous-time Takagi–Sugeno (T–S) fuzzy systems. Specifically, by using Lyapunov’s stability theory and the linear matrix inequality (LMI) technique, sufficient conditions for proving the local stability are provided in terms of single-parameter minimization problems subject to LMI constraints or eigenvalue problems, which are solvable via convex optimizations. The fuzzy Lyapunov functions (FLFs), expressed by the so-called multi-dimensional fuzzy summations, are employed to characterize invariant subsets of the DA as sublevel sets of the FLFs. To compute a larger inner estimate of the DA, an iterative LMI algorithm is also developed. Finally, illustrative examples show the efficacy of the approach.

Completeness of Lyapunov Abstraction

In this work, we continue our study on discrete abstractions of dynamical systems. To this end, we use a family of partitioning functions to generate an abstraction. The intersection of sub-level sets of the partitioning functions defines cells, which are regarded as discrete objects. The union of cells makes up the state space of the dynamical systems. Our construction gives rise to a combinatorial object - a timed automaton. We examine sound and complete abstractions. An abstraction is said to be sound when the flow of the time automata covers the flow lines of the dynamical systems. If the dynamics of the dynamical system and the time automaton are equivalent, the abstraction is complete. The commonly accepted paradigm for partitioning functions is that they ought to be transversal to the studied vector field. We show that there is no complete partitioning with transversal functions, even for particular dynamical systems whose critical sets are isolated critical points. Therefore, we allow the directional derivative along the vector field to be non-positive in this work. This considerably complicates the abstraction technique. For understanding dynamical systems, it is vital to study stable and unstable manifolds and their intersections. These objects appear naturally in this work. Indeed, we show that for an abstraction to be complete, the set of critical points of an abstraction function shall contain either the stable or unstable manifold of the dynamical system.

Compactness of Special Functions of Bounded Higher Variation

Abstract Given an open set Ω ⊂ Rm and n > 1, we introduce the new spaces GBnV(Ω) of Generalized functions of bounded higher variation and GSBnV(Ω) of Generalized special functions of bounded higher variation that generalize, respectively, the space BnV introduced by Jerrard and Soner in [43] and the corresponding SBnV space studied by De Lellis in [24]. In this class of spaces, which allow as in [43] the description of singularities of codimension n, the distributional jacobian Ju need not have finite mass: roughly speaking, finiteness of mass is not required for the (m−n)-dimensional part of Ju, but only finiteness of size. In the space GSBnV we are able to provide compactness of sublevel sets and lower semicontinuity of Mumford-Shah type functionals, in the same spirit of the codimension 1 theory [5,6].

An almost existence theorem for non-contractible periodic orbits in cotangent bundles

Assume M is a closed connected smooth manifold and H:T^*M->R a smooth proper function bounded from below. Suppose the sublevel set {H<d} contains the zero section and \alpha is a non-trivial homotopy class of free loops in M. Then for almost every s>=d the level set {H=s} carries a periodic orbit z of the Hamiltonian system (T^*M,\omega_0,H) representing \alpha. Examples show that the condition that {H<d} contains M is necessary and almost existence cannot be improved to everywhere existence.

Inner approximations of the region of attraction for polynomial dynamical systems

In a previous work we developed a convex infinite dimensional linear programming (LP) approach to approximating the region of attraction (ROA) of polynomial dynamical systems subject to compact basic semialgebraic state constraints. Finite dimensional relaxations to the infinite-dimensional LP lead to a truncated moment problem in the primal and a polynomial sum-of-squares problem in the dual. This primal-dual linear matrix inequality (LMI) problem can be solved numerically with standard semidefinite programming solvers, producing a hierarchy of outer (i.e. exterior) approximations of the ROA by polynomial sublevel sets, with a guarantee of almost uniform and set-wise convergence. In this companion paper, we show that our approach is flexible enough to be modified so as to generate a hierarchy of polynomial inner (i.e.\,interior) approximations of the ROA with similar convergence guarantees.

Elementary inequalities involving the roots of a polynomial with applications in harmonic analysis and number theory

We establish various inequalities relating the coefficients of a polynomial with the separation of its roots. Applications are given to oscillatory integrals and sublevel sets in euclidean harmonic analysis as well as exponential sums and polynomial congruences in number theory. These applications depend on precise structural statements of sublevel sets for polynomials with coefficients in a general field and these in turn give sharpened versions of classical results of Hua as well as Loxton and Smith regarding polynomial congruences.

A new class of Lyapunov functions for the constrained stabilization of linear systems

The constrained stabilization of linear uncertain systems is investigated via the set-theoretic framework of control Lyapunov R-functions. A novel composition rule allows the design of a composite control Lyapunov function with external level set that exactly shapes the maximal controlled invariant set and inner sublevel sets arbitrarily close to any choice of smooth ones, generalizing both polyhedral and truncated ellipsoidal control Lyapunov functions. The feasibility test of the proposed smooth control Lyapunov functions can be cast into matrix inequalities conditions. The constrained linear quadratic control is addressed as an application.

Complete abstractions of dynamical systems by timed automata

This paper addresses the generation of complete abstractions of polynomial dynamical systems by timed automata. For the proposed abstraction, the state space is divided into cells by sublevel sets of functions. We identify a relation between these functions and their directional derivatives along the vector field, which allows the generation of a complete abstraction.To compute the functions that define the subdivision of the state space in an algorithm, we formulate a sum of squares optimization problem. This optimization problem finds the best subdivisioning functions, with respect to the ability to approximate the dynamical system, in a subset of admissible subdivisioning functions.

Improvement sets and vector optimization

In this paper we focus on minimal points in linear spaces and minimal solutions of vector optimization problems, where the preference relation is defined via an improvement set E. To be precise, we extend the notion of E-optimal point due to Chicco et al. in [4] to a general (non-necessarily Pareto) quasi ordered linear space and we study its properties. In particular, we relate the notion of improvement set with other similar concepts of the literature and we characterize it by means of sublevel sets of scalar functions. Moreover, we obtain necessary and sufficient conditions for E-optimal solutions of vector optimization problems through scalarization processes by assuming convexity assumptions and also in the general (nonconvex) case. By applying the obtained results to certain improvement sets we generalize well-known results of the literature referred to efficient, weak efficient and approximate efficient solutions of vector optimization problems.

On the global solvability for overdetermined systems

We consider a class of systems of two smooth vector fields on the 3-torus associated to a closed 1-form. The global solvability of such systems is completely determined by the connectedness of the sublevel and superlevel sets of a primitive of this 1-form in the minimal covering. The main point of this talk is addressing the case when the 1-form is non-exact and its periods are rationally dependent. This is a work in collaboration with A. Bergamasco, A. Kirilov, and W. Nunes.

On a time-depending Monge–Ampère type equation

In this paper, we prove a comparison result between a solution u(x,t), x∈Ω⊂R2, t∈(0,T), of a time depending equation involving the Monge–Ampère operator in the plane and the solution of a conveniently symmetrized parabolic equation. To this aim, we prove a derivation formula for the integral of a smooth function g(x,t) over sublevel sets of u, {x∈Ω:u(x,t)<ϑ}, ϑ∈R, having the same perimeter in R2.

Subdifferential properties for a class of minimal time functions with moving target sets in normed spaces

In a normed vector space, we study the minimal time function determined by a moving target set and a differential inclusion, where the set-valued mapping involved has constant values of a bounded closed convex set U. After establishing a characterization of ϵ-subdifferential of the minimal time function, we obtain that the limiting subdifferential of the minimal time function is representable by virtue of the corresponding normal cones of sublevel sets of the function and level or sublevel sets of the support function of U. The known results require the set U to have the origin as an interior point and the target set is a fixed set.

Nonsmooth optimization via quasi-Newton methods

We investigate the behavior of quasi-Newton algorithms applied to minimize a nonsmooth function f, not necessarily convex. We introduce an inexact line search that generates a sequence of nested intervals containing a set of points of nonzero measure that satisfy the Armijo and Wolfe conditions if f is absolutely continuous along the line. Furthermore, the line search is guaranteed to terminate if f is semi-algebraic. It seems quite difficult to establish a convergence theorem for quasi-Newton methods applied to such general classes of functions, so we give a careful analysis of a special but illuminating case, the Euclidean norm, in one variable using the inexact line search and in two variables assuming that the line search is exact. In practice, we find that when f is locally Lipschitz and semi-algebraic with bounded sublevel sets, the BFGS (Broyden–Fletcher–Goldfarb–Shanno) method with the inexact line search almost always generates sequences whose cluster points are Clarke stationary and with function values converging R-linearly to a Clarke stationary value. We give references documenting the successful use of BFGS in a variety of nonsmooth applications, particularly the design of low-order controllers for linear dynamical systems. We conclude with a challenging open question.

Globally solvable systems of complex vector fields

We consider a class of involutive systems of n smooth vector fields on the n+1 dimensional torus. We obtain a complete characterization for the global solvability of this class in terms of Liouville forms and of the connectedness of all sublevel and superlevel sets of the primitive of a certain 1-form in the minimal covering space.

Characterizing Optimal Multiport Matching Transformers

Multiport transformers offer enhanced matching capabilities for multiport loads. These transformers are analyzed to characterize wideband matching designs. An eigenvalue decomposition of the multiport load produces a geometric description of the sublevel sets of the matching transformers and identifies those frequency bands where multiport transformers are effective. A multiport antenna provides a concrete design example.