Noncompact Sets Research Articles

Non-convex semi-infinite min-max optimization with noncompact sets

In this paper, first we study the non-convex sup-type functions with noncompact sets. Under quite mild conditions, the expressions of its derivative and subderivative along arbitrary direction are given. Furthermore, the structure of its subdifferential is characterized completely. Then, using these results, we establish first-order optimality conditions for semi-infinite min-max optimization problems. These results generalize and improve the corresponding results in the relevant literatures.

Limiting distributions for countable state topological Markov chains with holes

We study the dynamics of countable state topological Markov chains with holes, where the hole is a countable union of 1-cylinders. For a large class of positive recurrent potentials and under natural assumptions on the surviving dynamics, we prove the existence of a limiting conditionally invariant distribution, which is the unique limit of regular densities under the renormalized dynamics conditioned on non-escape. We also prove the existence of a Gibbs measure on the survivor set, the set of points that never enter the hole, which is an equilibrium measure for the punctured potential of the open system. We prove that the Gurevic pressure on the survivor set equals the exponential escape rate from the open system. These results extend to the non-compact setting results previously available for finite state topological Markov chains.

Local spectral gap in simple Lie groups and applications

We introduce a novel notion of local spectral gap for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action $$\Gamma \curvearrowright G$$ , whenever $$\Gamma $$ is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group G. This extends to the non-compact setting works of Bourgain and Gamburd (Invent Math 171:83–121, 2008; J Eur Math Soc (JEMS) 14:1455–1511, 2012), and Benoist and de Saxce (Invent Math 205:337–361, 2016). We present several applications to the Banach–Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on G. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique $$\Gamma $$ -invariant finitely additive measure defined on all bounded measurable subsets of G.

A local curvature estimate for the Ricci flow

We show that the norm of the Riemann curvature tensor of any smooth solution to the Ricci flow can be explicitly estimated in terms of its initial values on a given ball, a local uniform bound on the Ricci tensor, and the elapsed time. This provides a new, direct proof of a result of Šesǔm, which asserts that the curvature of a solution on a compact manifold cannot blow up while the Ricci curvature remains bounded, and extends its conclusions to the noncompact setting. We also prove that the Ricci curvature must blow up at least linearly along a subsequence at a finite time singularity.

Existence of equilibrium pair in non-compact E-proximity settings

In this paper, using the existence theorems of best proximity points and maximal elements in non-compact settings, we will prove a basic existence theorem of equilibrium pair for a non-compact free 1-person game which generalizes both xed point theorems and equilibrium existence theorems in non-compact E-proximity settings. Mathematics Subject Classication: 47H10, 91A06

Inviscid Burgers equation with random kick forcing in noncompact setting

We develop ergodic theory of the inviscid Burgers equation with random kick forcing in noncompact setting. The results are parallel to those in our recent work on the Burgers equation with Poissonian forcing. However, the analysis based on the study of one-sided minimizers of the relevant action is different. In contrast with previous work, finite time coalescence of the minimizers does not hold, and hyperbolicity (exponential convergence of minimizers in reverse time) is not known. In order to establish a One Force --- One Solution principle on each ergodic component, we use an extremely soft method to prove a weakened hyperbolicity property and to construct Busemann functions along appropriate subsequences.

Generalized bi-quasi-variational inequalities for quasi-pseudo-monotone type II operators in non-compact settings

In this paper, we introduce a new class of generalized bi-quasi-variational inequalities for quasipseudo- monotone type II operators in non-compact settings of locally convex Hausdorff topological vector spaces and show the existence results of solutions for generalized bi-quasi-variational inequalities. Our results improve, extend and generalized the corresponding results given by some authors

Cooperative Global Output Regulation of Second-Order Nonlinear Multi-Agent Systems With Unknown Control Direction

This technical note addresses the cooperative global output regulation problem for second-order nonlinear multi-agent systems without a priori knowledge of the control direction. We will propose both the graph-dependent and graph-independent distributed controllers based on the combination of the internal model, the Nussbaum-type dynamic gain, and the adaptive control technique. Comparing with some existing results, our novel design can handle the multi-agent system with not only the arbitrary and non-identical unknown control direction but also the uncertain parameter that belongs to any unknown and non-compact set.

Quantization coefficients in infinite systems

We investigate quantization coefficients for self-similar probability measures \mu on limit sets which are generated by systems S of infinitely many contractive similarities and by probabilistic vectors. The theory of quantization coefficients for infinite systems has significant differences from the finite case. One of these differences is the lack of finite maximal antichains, and the fact that the set of contraction ratios has zero infimum; another difference resides in the specific geometry of the non-compact limit set J of S. We prove that, for each r \in (0,1), there exists a unique positive number \kappa_r, so that for arbitrary \kappa < \kappa_r < \kappa', the \kappa-dimensional lower quantization coefficient of order r of \mu is positive, and we also give estimates for the \kappa'-dimensional upper quantization coefficient of order r of \mu. In particular, it follows that the quantization dimension of order r of \mu exists, and it is equal to \kappa_r. The above results allow then to estimate the asymptotic errors of approximating the measure \mu in the L_r-Kantorovich-Wasserstein metric, with discrete measures supported on finitely many points.

Design of Optimal Scan Tree Based on Compact Test Patterns for Test Time Reduction

Scan tree architecture has been proposed to reduce the test application time of full scan chain by placing multiple scan cells in parallel. Most existing techniques rely on non-compact test pattern sets to construct the scan tree. However, they produce inefficient scan tree when highly compact test sets with few don't cares are used. In this paper, the depth of the scan tree based on approximate compatibility relation for completely specified test data set is analyzed probabilistically by modeling its construction as a vertex coloring problem. The upper bound of edges-per-vertex is computed and demonstrated to be a prime factor that limits the efficiency of scan tree construction based on both compatible and approximately compatible test data between two flip-flops. Inverse compatibility and aggressive approximate compatibility are then proposed to increase the edges-per-vertex for vertex coloring. The Q′-SD connection between two adjacent scan cells is exploited to implement the inverse compatibility with no cost or timing impact. To maintain the fault coverage, the missing faults under the tree scan mode can be detected by switching the same base architecture into the linear scan mode with negligible hardware overhead as shown by the experimental results on ISCAS89, ISCAS99 and LGSynth93 benchmark circuits. On average, the scan tree generated by our method reduces the test time of the full scan chain by 56.65 percent, and that of the scan tree designed by the approximate compatibility method by 39.18 percent under the same compact test sets.

Non-compactness and infinite number of conformal initial data sets in high dimensions

On any closed Riemannian manifold of dimension greater than 7, we construct examples of background physical coefficients for which the Einstein–Lichnerowicz equation possesses a non-compact set of positive solutions. This yields in particular the existence of an infinite number of positive solutions in such cases.

Quantization commutes with reduction in the non-compact setting: the case of holomorphic discrete series

In this paper we show that the multiplicities of holomorphic discrete series representations relative to reductive subgroups satisfy the credo “quantization commutes with reduction”.

Continuity of Minima: Local Results

Berge’s maximum theorem is an important statement in set-valued analysis that has significant applications in mathematical economics, operations research, and control. For a minimization problem for a continuous function of two variables and continuous compact-valued set-valued mapping defining feasible sets, this theorem states continuity of the value function and upper semi-continuity of the solution multifunction. One of the main limitations of this theorem is that the set-valued mapping is compact-valued. Recently the authors of this paper and their coauthors generalized Berge’s maximum theorem to set-valued mappings that may not be compact-valued. Here we formulate and prove the local Berge’s maximum theorem for possibly noncompact feasible sets and show that it is more general than the recently established Berge’s maximum theorem for possibly noncompact feasible sets and than the known formulations of the local Berge’s maximum theorem.

Generalized Quasi-Variational Inequalities for Pseudo-Monotone Type III and Strongly Pseudo-Monotone Type III Operators on Non-Compact Sets

In this paper, the authors prove some existence results of solutions for a new class of generalized quasi-variational inequalities (GQVI) for pseudo-monotone type III operators and strongly pseudo-monotone type III operators defined on non-compact sets in locally convex Hausdorff topological vector spaces. In obtaining these results on GQVI for pseudo-monotone type III operators, we shall use Chowdhury and Tan’s generalized version [1] of Ky Fan’s minimax inequality [2] as the main tool.

Existence Results for New Weak and Strong Mixed Vector Equilibrium Problems on Noncompact Domain

We introduce and consider two new mixed vector equilibrium problems, that is, a new weak mixed vector equilibrium problem and a new strong mixed vector equilibrium problem which are combinations of certain vector equilibrium problems, and vector variational inequality problems. We prove existence results for the problems in noncompact setting.

Semidefinite Representations of Noncompact Convex Sets

We consider the problem of the semidefinite representation of a class of noncompact basic semialgebraic sets. We introduce the conditions of pointedness and closedness at infinity of a semialgebraic set and show that under these conditions our modified hierarchies of nested theta bodies and Lasserre's relaxations converge to the closure of the convex hull of $S$. Moreover, if the Putinar-Prestel's Bounded Degree Representation (PP-BDR) property is satisfied, our theta body and Lasserre's relaxation are exact when the order is large enough; if the PP-BDR property does not hold, our hierarchies converge uniformly to the closure of the convex hull of $S$ restricted to every fixed ball centered at the origin. We illustrate through a set of examples that the conditions of pointedness and closedness are essential to ensure the convergence. Finally, we provide some strategies to deal with cases where the conditions of pointedness and closedness are violated.

On affine maps on non-compact convex sets and some characterizations of finite-dimensional solid ellipsoids

Convex geometry has recently attracted great attention as a framework to formulate general probabilistic theories. In this framework, convex sets and affine maps represent the state spaces of physical systems and the possible dynamics, respectively. In the first part of this paper, we present a result on separation of simplices and balls (up to affine equivalence) among all compact convex sets in two- and three-dimensional Euclidean spaces, which focuses on the set of extreme points and the action of affine transformations on it. Regarding the above-mentioned axiomatization of quantum physics, our result corresponds to the case of simplest (2-level) quantum system. We also discuss a possible extension to higher dimensions. In the second part, towards generalizations of the framework of general probabilistic theories and several existing results including ones in the first part from the case of compact and finite-dimensional physical systems as in most of the literature to more general cases, we study some fundamental properties of convex sets and affine maps that are relevant to the above subject.

Satellite cluster flight using on-off cyclic control

Nano-satellite clusters and disaggregated satellites are new concepts in the realm of distributed satellite systems, which require complex cluster management – mainly regulating the maximal and minimal inter-satellite distances on time scales of years – while utilizing simple on-off propulsion systems. The simple actuators and long time scales require judicious astrodynamical modeling coupled with specialized orbit control. This paper offers a satellite cluster orbit control law which works for long time scales in a perturbed environment while utilizing fixed-magnitude thrusters. The main idea is to design a distributed controller which balances the fuel consumption among the satellites, thus mitigating the effect of differential drag perturbations. The underlying methodology utilizes a cyclic control algorithm based on a mean orbital elements feedback. Stability properties of the closed-loop cyclic control system do not adhere to the classical Lyapunov stability theory, so an effort is made to define and implement a suitable stability theory of noncompact equilibria sets. A state selection scheme is proposed for efficiently establishing a low Earth orbit cluster. Several simulations, including a real mission study, and several comparative investigations, are performed to show the strengths of the proposed control law.

A Variational Formula for the Lyapunov Exponent of Brownian Motion in Stationary Ergodic Potential

We establish a variational formula for the exponential decay rate of the Green function of Brownian motion evolving in a random stationary and ergodic nonnegative potential. Such a variational formula is established by Schroeder (1988) for periodic potentials and is generalised in the present article to a non-compact setting. We show exponential decay of the Green function implicitly. This formula for the Lyapunov exponent has several direct implications. It allows to compare the inuence of a random potential to the inuence of the averaged potential. It also leads to a variational expression for the quenched free energy.

Slow Entropy for Noncompact Sets and Variational Principle

This paper defines and discusses the dimension notion of topological slow entropy of any subset for \({{\mathbb {Z}}}^d-\)actions. Also, the notion of measure-theoretic slow entropy for \({\mathbb {Z}}^d-\)actions is presented, which is modified from Brin and Katok (Geometric Dynamics, Springer, Berlin 1983). Relations between Bowen topological entropy Bowen (Trans Am Math, 184:125–136, 1973), and topological slow entropy are studied in this paper, and several examples of the topological slow entropy in a symbolic system are given. Specifically, a variational principle is proved.