Fuzzy Continuity of Almost Linear Operators

2013 ◽  
Vol 3 (1) ◽  
pp. 40-50
Author(s):  
Mark Burgin
Keyword(s):  
Classical Result ◽  
Linear Operators ◽  
Vector Spaces ◽  
Future Research ◽  
Operator Equations ◽  
Open Problems ◽  
Direct Corollary ◽  
Theory Of Information ◽  
Normed Vector ◽  
Neoclassical Analysis

In this paper, the author studies relations between fuzzy continuity and boundedness of approximately linear operators in the context of neoclassical analysis. The main result of this paper (Theorem 1) demonstrates that for approximately linear operators, fuzzy continuity is equivalent to boundedness when the continuity defect (or measure of discontinuity) is sufficiently small. The classical result that describes continuity of linear operators becomes a direct corollary of this theorem. Applying Theorem 1, we demonstrate (Theorem 2) that for linear operators in normed vector spaces, fuzzy continuity coincides with continuity when the continuity defect is sufficiently small, i.e., when it is less than one. Results are oriented at applications in physics, theory of information and other fields where operator equations play an important role. Several open problems and directions for future research are considered at the end of the paper.

scholarly journals Climate Change Sustainability: From Bargaining to Cooperative Balanced Approach

Games ◽  
2021 ◽  
Vol 12 (2) ◽  
pp. 45
Author(s):  
Tiziana Ciano ◽  
Massimiliano Ferrara ◽  
Mariangela Gangemi ◽  
Domenica Stefania Merenda ◽  
Bruno Antonio Pansera
Keyword(s):  
Climate Change ◽  
Cooperative Game Theory ◽  
Research Field ◽  
Future Research ◽  
Open Problems ◽  
Cooperative Approach ◽  
Change Dynamics ◽  
Bargaining Solutions ◽  
Balanced Approach ◽  
Competitive Bargaining

This work aims to provide different perspectives on the relationships between cooperative game theory and the research field concerning climate change dynamics. New results are obtained in the framework of competitive bargaining solutions and related issues, moving from a cooperative approach to a competitive one. Furthermore, the dynamics of balanced and super-balanced games are exposed, with particular reference to coalitions. Some open problems are presented to aid future research in this area.

Hierarchical Reinforcement Learning

2021 ◽  
Vol 54 (5) ◽  
pp. 1-35
Author(s):  
Shubham Pateria ◽  
Budhitama Subagdja ◽  
Ah-hwee Tan ◽  
Chai Quek
Keyword(s):  
Reinforcement Learning ◽  
Future Research ◽  
Comprehensive Overview ◽  
Open Problems ◽  
Practical Applications ◽  
Hierarchical Reinforcement Learning ◽  
The Past ◽  
Agent Learning ◽  
Multi Agent ◽  
Supplementary Material

Hierarchical Reinforcement Learning (HRL) enables autonomous decomposition of challenging long-horizon decision-making tasks into simpler subtasks. During the past years, the landscape of HRL research has grown profoundly, resulting in copious approaches. A comprehensive overview of this vast landscape is necessary to study HRL in an organized manner. We provide a survey of the diverse HRL approaches concerning the challenges of learning hierarchical policies, subtask discovery, transfer learning, and multi-agent learning using HRL. The survey is presented according to a novel taxonomy of the approaches. Based on the survey, a set of important open problems is proposed to motivate the future research in HRL. Furthermore, we outline a few suitable task domains for evaluating the HRL approaches and a few interesting examples of the practical applications of HRL in the Supplementary Material.

MetaLearning with Graph Neural Networks

2021 ◽  
Vol 23 (2) ◽  
pp. 13-22
Author(s):  
Debmalya Mandal ◽  
Sourav Medya ◽  
Brian Uzzi ◽  
Charu Aggarwal
Keyword(s):  
Neural Networks ◽  
Deep Neural Networks ◽  
Future Research ◽  
Open Problems ◽  
Research Directions ◽  
Graph Problems ◽  
Meta Learning ◽  
Comprehensive Survey ◽  
Future Research Directions ◽  
Graph Neural Networks

Graph Neural Networks (GNNs), a generalization of deep neural networks on graph data have been widely used in various domains, ranging from drug discovery to recommender systems. However, GNNs on such applications are limited when there are few available samples. Meta-learning has been an important framework to address the lack of samples in machine learning, and in recent years, researchers have started to apply meta-learning to GNNs. In this work, we provide a comprehensive survey of different metalearning approaches involving GNNs on various graph problems showing the power of using these two approaches together. We categorize the literature based on proposed architectures, shared representations, and applications. Finally, we discuss several exciting future research directions and open problems.

Functionals of Bounded Frechet Variation

1949 ◽  
Vol 1 (2) ◽  
pp. 153-165 ◽  
Author(s):  
Marston Morse ◽  
William Transue
Keyword(s):  
Representation Theory ◽  
Spectral Theory ◽  
General Type ◽  
Integrodifferential Equations ◽  
Vector Spaces ◽  
Variational Theory ◽  
Characteristic Equations ◽  
Special Cases ◽  
Jacobi Equations ◽  
Normed Vector

In a series of papers which will follow this paper the authors will present a theory of functionals which are bilinear over a product A × B of two normed vector spaces A and B. This theory will include a representation theory, a variational theory, and a spectral theory. The associated characteristic equations will include as special cases the Jacobi equations of the classical variational theory when n = 1, and self-adjoint integrodifferential equations of very general type. The bilinear theory is oriented by the needs of non-linear and non-bilinear analysis in the large.

Control of flexible manipulators: A survey

Robotica ◽  
2004 ◽  
Vol 22 (5) ◽  
pp. 533-545 ◽  
Author(s):  
M. Benosman ◽  
G. Le Vey
Keyword(s):  
Linear Models ◽  
Control Strategies ◽  
Research Area ◽  
Future Research ◽  
Complex Nature ◽  
Open Problems ◽  
Complex Models ◽  
The Many ◽  
The One ◽  
Flexible Arms

A survey of the field of control for flexible multi-link robots is presented. This research area has drawn great attention during the last two decades, and seems to be somewhat less “attractive” now, due to the many satisfactory results already obtained, but also because of the complex nature of the remaining open problems. Thus it seems that the time has come to try to deliver a sort of “state of the art” on this subject, although an exhaustive one is out of scope here, because of the great amount of publications. Instead, we survey the most salient progresses – in our opinion – approximately during the last decade, that are representative of the essential different ideas in the field. We proceed along with the exposition of material coming from about 119 included references. We do not pretend to deeply present each of the methods quoted hereafter; however, our goal is to briefly introduce most of the existing methods and to refer the interested reader to more detailed presentations for each scheme. To begin with, a now well-established classification of the flexible arms control goals is given. It is followed by a presentation of different control strategies, indicating in each case whether the approach deals with the one-link case, which can be successfully treated via linear models, or with the multi-link case which necessitates nonlinear, more complex, models. Some possible issues for future research are given in conclusion.

scholarly journals A holistic review on advanced bi-directional EV charging control algorithms

2021 ◽  
Vol 1 (1) ◽  
pp. 78-88
Author(s):  
Xiaoying Tang ◽  
Chenxi Sun ◽  
Suzhi Bi ◽  
Shuoyao Wang ◽  
Angela Yingjun Zhang
Keyword(s):  
Electric Vehicles ◽  
Electricity Market ◽  
Control Algorithms ◽  
Future Research ◽  
Scheduling Problems ◽  
Time Data ◽  
Open Problems ◽  
Holistic Review ◽  
Charging Control ◽  
Ev Charging

The rapid growth of electric vehicles (EVs) has promised a next-generation transportation system with reduced carbon emission. The fast development of EVs and charging facilities is driving the evolution of Internet of Vehicles (IoV) to Internet of Electric Vehicles (IoEV). IoEV benefits from both smart grid and Internet of Things (IoT) technologies which provide advanced bi-directional charging services and real-time data processing capability, respectively. The major design challenges of the IoEV charging control lie in the randomness of charging events and the mobility of EVs. In this article, we present a holistic review on advanced bi-directional EV charging control algorithms. For Grid-to-Vehicle (G2V), we introduce the charging control problem in two scenarios: 1) Operation of a single charging station and 2) Operation of multiple charging stations in coupled transportation and power networks. For Vehicle-to-Grid (V2G), we discuss how EVs can perform energy trading in the electricity market and provide ancillary services to the power grid. Besides, a case study is provided to illustrate the economic benefit of the joint optimization of routing and charging scheduling of multiple EVs in the IoEV. Last but not the least, we will highlight some open problems and future research directions of charging scheduling problems for IoEVs.

scholarly journals Subdifferential Properties of Minimal Time Functions Associated with Set-Valued Mappings with Closed Convex Graphs in Hausdorff Topological Vector Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Messaoud Bounkhel
Keyword(s):  
Optimization Problems ◽  
Time Function ◽  
Vector Spaces ◽  
Directional Derivatives ◽  
Minimal Time Function ◽  
Topological Vector Spaces ◽  
Minimal Time ◽  
Set Valued Mapping ◽  
Bounded Set ◽  
Normed Vector

For a set-valued mappingMdefined between two Hausdorff topological vector spacesEandFand with closed convex graph and for a given point(x,y)∈E×F, we study the minimal time function associated with the images ofMand a bounded setΩ⊂Fdefined by𝒯M,Ω(x,y):=inf{t≥0:M(x)∩(y+tΩ)≠∅}. We prove and extend various properties on directional derivatives and subdifferentials of𝒯M,Ωat those points of(x,y)∈E×F(both cases: points in the graphgph Mand points outside the graph). These results are used to prove, in terms of the minimal time function, various new characterizations of the convex tangent cone and the convex normal cone to the graph ofMat points insidegph Mand to the graph of the enlargement set-valued mapping at points outsidegph M. Our results extend many existing results, from Banach spaces and normed vector spaces to Hausdorff topological vector spaces (Bounkhel, 2012; Bounkhel and Thibault, 2002; Burke et al., 1992; He and Ng, 2006; and Jiang and He 2009). An application of the minimal time function𝒯M,Ωto the calmness property of perturbed optimization problems in Hausdorff topological vector spaces is given in the last section of the paper.

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