Explore the vast range of titles available.

In this study, we conduct a literature review on normed linear spaces whose strengths are between rotundity and uniform rotundity. In this discourse, we also explore inter-relationships and juxtapositions between the subjects under consideration. There has been some discussion on the extent to which the geometry of the factor spaces has an impact on the geometry of the product spaces, as well as the degree to which the quotient spaces and subspaces inherit the geometry of the space itself. A comprehensive review has been conducted on the applications of most of these rotundities to some fields within the realm of approximation theory. In addition, some open problems are enumerated in the paper.

In Optimal Recovery, the task of learning a function from observational data is tackled deterministically by adopting a worst-case perspective tied to an explicit model assumption made on the functions to be learned. Working in the framework of Hilbert spaces, this article considers a model assumption based on approximability. It also incorporates observational inaccuracies modeled via additive errors bounded in ℓ 2 . Earlier works have demonstrated that regularization provides algorithms that are optimal in this situation, but did not fully identify the desired hyperparameter. This article fills the gap in both a local scenario and a global scenario. In the local scenario, which amounts to the determination of Chebyshev centers, the semidefinite recipe of Beck and Eldar (legitimately valid in the complex setting only) is complemented by a more direct approach, with the proviso that the observational functionals have orthonormal representers. In the said approach, the desired parameter is the solution to an equation that can be resolved via standard methods. In the global scenario, where linear algorithms rule, the parameter elusive in the works of Micchelli et al. is found as the byproduct of a semidefinite program. Additionally and quite surprisingly, in case of observational functionals with orthonormal representers, it is established that any regularization parameter is optimal.

This paper is dedicated to an optimization problem. Let A , B ⊂ be compact convex sets. Consider the minimal number t 0 > 0 such that t 0 B covers A after a shift to a vector x 0 ∈ . The goal is to find t 0 and x 0 . In the special case of B being a unit ball centered at zero, x 0 and t 0 are known as the Chebyshev center and the Chebyshev radius of A . This paper focuses on the case in which A and B are defined with their black-box support functions. An algorithm for solving such problems efficiently is suggested. The algorithm has a superlinear convergence rate, and it can solve hundred-dimensional test problems in a reasonable time, but some additional conditions on A and B are required to guarantee the presence of convergence. Additionally, the behavior of the algorithm for a simple special case is investigated, which leads to a number of theoretical results. Perturbations of this special case are also studied.

We propose a convex quadratic programming (CQP) relaxation for multi-ball constrained quadratic optimization (MB). (CQP) is shown to be equivalent to semidefinite programming relaxation in the hard case. Based on (CQP), we propose an algorithm for solving (MB), which returns a solution of (MB) with an approximation bound independent of the number of constraints. The approximation algorithm is further extended to solve nonconvex quadratic optimization with more general constraints. As an application, we propose a standard quadratic programming relaxation for finding Chebyshev center of a general convex set with a guaranteed approximation bound.

For a compact Hausdorff space S , we prove that the closed unit ball of a closed linear subalgebra of the space of real-valued continuous functions on S , denoted by C ( S ), satisfies property- ( P 1 ) (the set-valued generalization of strong proximinality) for the non-empty closed bounded subsets of the bidual of C ( S ). Various stability results related to property- ( P 1 ) and semi-continuity properties of restricted Chebyshev-center maps are also established. As a consequence, we derive that if Y is a proximinal finite co-dimensional subspace of c 0 then the closed unit ball of Y satisfies property- ( P 1 ) for the non-empty closed bounded subsets of ℓ ∞ and the restricted Chebyshev-center map of the closed unit ball of Y is Hausdorff metric continuous on certain subclasses of the class of non-empty closed bounded subsets of ℓ ∞ .

In this paper we study closed subspaces of Banach spaces that admit relative Chebyshev centers for all bounded subsets of the space. We exhibit new classes of spaces which have this property and study stability results similar to the ones studied in the literature in the context of proximinal subspaces and Chebyshev centers. For the space of continuous functions on a compact set , we show that a closed subspace of finite codimension has relative Chebyshev centers for all bounded sets in if and only if it is a strongly proximinal subspace.

This self-contained introduction to the distributed control of robotic networks offers a distinctive blend of computer science and control theory. The book presents a broad set of tools for understanding coordination algorithms, determining their correctness, and assessing their complexity; and it analyzes various cooperative strategies for tasks such as consensus, rendezvous, connectivity maintenance, deployment, and boundary estimation. The unifying theme is a formal model for robotic networks that explicitly incorporates their communication, sensing, control, and processing capabilities--a model that in turn leads to a common formal language to describe and analyze coordination algorithms. Written for first- and second-year graduate students in control and robotics, the book will also be useful to researchers in control theory, robotics, distributed algorithms, and automata theory. The book provides explanations of the basic concepts and main results, as well as numerous examples and exercises. Self-contained exposition of graph-theoretic concepts, distributed algorithms, and complexity measures for processor networks with fixed interconnection topology and for robotic networks with position-dependent interconnection topology Detailed treatment of averaging and consensus algorithms interpreted as linear iterations on synchronous networks Introduction of geometric notions such as partitions, proximity graphs, and multicenter functions Detailed treatment of motion coordination algorithms for deployment, rendezvous, connectivity maintenance, and boundary estimation

We obtain formulae to calculate the asymptotic center and radius of bounded sequences in C 0 ( L ) spaces. We also study the existence of continuous selectors for the asymptotic center map in general Banach spaces. In Hilbert spaces, even a Hölder-type estimation is given.

We consider the problem of estimating a vector ${\\bf z}$ in the regression model $\\mbox{$\\mathcal{B}$} = {\\bf A} {\\bf z} + {\\bf w}$, where ${\\bf w}$ is an unknown but bounded noise. As in many regularization schemes, we assume that an upper bound on the norm of ${\\bf z}$ is available. To estimate ${\\bf z}$ we propose a relaxation of the Chebyshev center, which is the vector that minimizes the worst-case estimation error over all feasible vectors ${\\bf z}$. Relying on recent results regarding strong duality of nonconvex quadratic optimization problems with two quadratic constraints, we prove that in the complex domain our approach leads to the exact Chebyshev center. In the real domain, this strategy results in a \"pretty good\" approximation of the true Chebyshev center. As we show, our estimate can be viewed as a Tikhonov regularization with a special choice of parameter that can be found efficiently by solving a convex optimization problem with two variables or a semidefinite program with three variables, regardless of the problem size. When the norm constraint on ${\\bf z}$ is a Euclidean one, the problem reduces to a single-variable convex minimization problem. We then demonstrate via numerical examples that our estimator can outperform other conventional methods, such as least-squares and regularized least-squares, with respect to the estimation error. Finally, we extend our methodology to other feasible parameter sets, showing that the total least-squares (TLS) and regularized TLS can be obtained as special cases of our general approach.

Background Size and shape of the treatment zone after Irreversible electroporation (IRE) can be difficult to depict due to the use of multiple applicators with complex spatial configuration. Exact geometrical definition of the treatment zone, however, is mandatory for acute treatment control since incomplete tumor coverage results in limited oncological outcome. In this study, the “Chebyshev Center Concept” was introduced for CT 3d rendering to assess size and position of the maximum treatable tumor at a specific safety margin. Methods In seven pig livers, three different IRE protocols were applied to create treatment zones of different size and shape: Protocol 1 (n = 5 IREs), Protocol 2 (n = 5 IREs), and Protocol 3 (n = 5 IREs). Contrast-enhanced CT was used to assess the treatment zones. Technique A consisted of a semi-automated software prototype for CT 3d rendering with the “Chebyshev Center Concept” implemented (the “Chebyshev Center” is the center of the largest inscribed sphere within the treatment zone) with automated definition of parameters for size, shape and position. Technique B consisted of standard CT 3d analysis with manual definition of the same parameters but position. Results For Protocol 1 and 2, short diameter of the treatment zone and diameter of the largest inscribed sphere within the treatment zone were not significantly different between Technique A and B. For Protocol 3, short diameter of the treatment zone and diameter of the largest inscribed sphere within the treatment zone were significantly smaller for Technique A compared with Technique B (41.1 ± 13.1 mm versus 53.8 ± 1.1 mm and 39.0 ± 8.4 mm versus 53.8 ± 1.1 mm; p < 0.05 and p < 0.01). For Protocol 1, 2 and 3, sphericity of the treatment zone was significantly larger for Technique A compared with B. Conclusions Regarding size and shape of the treatment zone after IRE, CT 3d rendering with the “Chebyshev Center Concept” implemented provides significantly different results compared with standard CT 3d analysis. Since the latter overestimates the size of the treatment zone, the “Chebyshev Center Concept” could be used for a more objective acute treatment control.