Structural aspects of the non-uniformly continuous functions and the unbounded functions within C(X)

We prove in this paper that if a metric space supports a real continuous function which is not uniformly continuous then, under appropriate mild assumptions, there exists in fact a plethora of such functions, in both topological and algebraical senses. Corresponding results are also obtained concerning unbounded continuous functions on a non-compact metrizable space.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Economía y Competitividad (MINECO). Españ

Nowhere hölderian functions and Pringsheim singular functions in the disc algebra

We prove the existence of dense linear subspaces, of infinitely generated subalgebras and of infinite dimensional Banach spaces in the disc algebra all of whose nonzero members are not α-h¨olderian at any point of the unit circle for any α > 0. This completes the recently established result of topological genericity of this kind of functions, as well as the corresponding lineability statements about functions that are nowhere differentiable at the boundary. Topological and algebraic genericity is also studied for the family of boundary-smooth holomorphic functions that are Pringsheim singular at any point of the unit circle.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Economía y Competitividad (MINECO). Españ

Learning and Management for Internet-of-Things: Accounting for Adaptivity and Scalability

Internet-of-Things (IoT) envisions an intelligent infrastructure of networked smart devices offering task-specific monitoring and control services. The unique features of IoT include extreme heterogeneity, massive number of devices, and unpredictable dynamics partially due to human interaction. These call for foundational innovations in network design and management. Ideally, it should allow efficient adaptation to changing environments, and low-cost implementation scalable to massive number of devices, subject to stringent latency constraints. To this end, the overarching goal of this paper is to outline a unified framework for online learning and management policies in IoT through joint advances in communication, networking, learning, and optimization. From the network architecture vantage point, the unified framework leverages a promising fog architecture that enables smart devices to have proximity access to cloud functionalities at the network edge, along the cloud-to-things continuum. From the algorithmic perspective, key innovations target online approaches adaptive to different degrees of nonstationarity in IoT dynamics, and their scalable model-free implementation under limited feedback that motivates blind or bandit approaches. The proposed framework aspires to offer a stepping stone that leads to systematic designs and analysis of task-specific learning and management schemes for IoT, along with a host of new research directions to build on.Comment: Submitted on June 15 to Proceeding of IEEE Special Issue on Adaptive and Scalable Communication Network

A Combined Preconditioning Strategy for Nonsymmetric Systems

We present and analyze a class of nonsymmetric preconditioners within a normal (weighted least-squares) matrix form for use in GMRES to solve nonsymmetric matrix problems that typically arise in finite element discretizations. An example of the additive Schwarz method applied to nonsymmetric but definite matrices is presented for which the abstract assumptions are verified. A variable preconditioner, combining the original nonsymmetric one and a weighted least-squares version of it, is shown to be convergent and provides a viable strategy for using nonsymmetric preconditioners in practice. Numerical results are included to assess the theory and the performance of the proposed preconditioners.Comment: 26 pages, 3 figure

Non-Lipschitz differentiable functions on slit domains

It is proved the existence of large algebraic structures –including large vector subspaces or infinitely generated free algebras– inside the family of non-Lipschitz differentiable real functions with bounded gradient defined on special non-convex plane domains. In particular, this yields that there are many differentiable functions on plane domains that do not satisfy the Mean Value Theorem.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Economía y Competitividad (MINECO). Españ

Large algebras of singular functions vanishing on prescribed sets

In this paper, the non-vacuousness of the family of all nowhere analytic infinitely differentiable functions on the real line vanishing on a prescribed set Z is characterized in terms of Z. In this case, large algebraic structures are found inside such family. The results obtained complete or extend a number of previous ones by several authors.Comment: 11 page