Network Theory and Applications – serie

Following the pattern of the Internet growth in popularity, started in the early 1990s, the current unprecedented expansion of wireless technology promises to have an even greater effect on how people communicate and interact, with considerable socio-economic impact all over the world. The driving force behind this growth is the remarkable progress in component miniaturization, integration, and also devel- ments in waveforms, coding, and communication protocols. Besides established infrastructurebased wireless networks (cellular, WLAN, sat- lite) ad-hoc wireless networks emerge as a new platform for distributed applications and for personal communication in scenarios where deploying infrastructure is not feasible. In ad-hoc wireless networks, each node is capable of forwarding packets on behalf of other nodes, so that multi-hop paths provide end-to-end connectivity. The increased flexibility and mobility of ad-hoc wireless networks are favored for appli- tions in law enforcement, homeland defense and military. In a world where wireless networks become increasingly interoperable with each other and with the high-speed wired Internet, personal communication systems will transform into universal terminals with instant access to variate content and able of handle demanding tasks, such as multimedia and real-time video. With users roaming between networks, and with wide variation in wireless link quality even in a single domain, the communications terminal must continue to provide a level of Quality of Service that is acceptable to the user and conforms to a contracted Service Level Agreement.

Both authors have taught the course of “Distributed Systems” for many years in the respective schools. During the teaching, we feel strongly that “Distributed systems” have evolved from traditional “LAN” based distributed systems towards “Internet based” systems. Although there exist many excellent textbooks on this topic, because of the fast development of distributed systems and network programming/protocols, we have difficulty in finding an appropriate textbook for the course of “distributed systems” with orientation to the requirement of the undergraduate level study for today’s distributed technology. Specifically, from - to-date concepts, algorithms, and models to implementations for both distributed system designs and application programming. Thus the philosophy behind this book is to integrate the concepts, algorithm designs and implementations of distributed systems based on network programming. After using several materials of other textbooks and research books, we found that many texts treat the distributed systems with separation of concepts, algorithm design and network programming and it is very difficult for students to map the concepts of distributed systems to the algorithm design, prototyping and implementations. This book intends to enable readers, especially postgraduates and senior undergraduate level, to study up-to-date concepts, algorithms and network programming skills for building modern distributed systems. It enables students not only to master the concepts of distributed network system but also to readily use the material introduced into implementation practices.

The huge bandwidth of optical fibre was recognized back in the 1970s during the early development of fibre optics technology. Between 1979 and 1999, the capacity of experimental and deployed systems has been increasing at a rate of 100-fold each decade - a rate exceeding the increase of integrated circuit speeds. Today, optical communication in the public communication networks has developed from the status of a curiosity into being the dominant technology. Various challenges arising from the deployment of wavelength division multiplexing (WDM) have attracted effort from many researchers. Indeed, optical networking has been a fertile ground for both theoretical research and experimental studies. This monograph presents contributions from the author's past and ongoing research in the field of optical networking. The ideas presented focus more on graph-theoretical and algorithmic aspects of optical networks.

These days, the nature of services and the volume of demand in the telecommu­ nication industry is changing radically, with the replacement of analog transmis­ sion and traditional copper cables by digital technology and fiber optic transmis­ sion equipment. Moreover, we see an increasing competition among providers of telecommunication services, and the development of a broad range of new services for users, combining voice, data, graphics and video. Telecommunication network planning has thus become an important problem area for developing and applying optimization models. Telephone companies have initiated extensive modeling and planning efforts to expand and upgrade their transmission facilities, which are, for most national telecommunication networks, divided in three main levels (see Balakrishnan et al. [5]), namely, l. the long-distance or backbone network that typically connects city pairs through gateway nodes; 2. the inter-office or switching center network within each city, that interconnects switching centers in different subdivisions (clusters of customers) and provides access to the gateway(s) node(s); 1 2 DESIGN OF SURVNABLE NETWORKS WITH BOUNDED RINGS 3. the local access network that connects individual subscribers belonging to a cluster to the corresponding switching center. These three levels differ in several ways including their design criteria. Ideally, the design of a telecommunication network should simultaneously account for these three levels. However, to simplify the planning task, the overall planning problem is decomposed by considering each level separately.

Combinatorics and Matrix Theory have a symbiotic, or mutually beneficial, relationship. This relationship is discussed in my paper The symbiotic relationship of combinatorics and matrix theoryl where I attempted to justify this description. One could say that a more detailed justification was given in my book with H. J. Ryser entitled Combinatorial Matrix Theon? where an attempt was made to give a broad picture of the use of combinatorial ideas in matrix theory and the use of matrix theory in proving theorems which, at least on the surface, are combinatorial in nature. In the book by Liu and Lai, this picture is enlarged and expanded to include recent developments and contributions of Chinese mathematicians, many of which have not been readily available to those of us who are unfamiliar with Chinese journals. Necessarily, there is some overlap with the book Combinatorial Matrix Theory. Some of the additional topics include: spectra of graphs, eulerian graph problems, Shannon capacity, generalized inverses of Boolean matrices, matrix rearrangements, and matrix completions. A topic to which many Chinese mathematicians have made substantial contributions is the combinatorial analysis of powers of nonnegative matrices, and a large chapter is devoted to this topic. This book should be a valuable resource for mathematicians working in the area of combinatorial matrix theory. Richard A. Brualdi University of Wisconsin - Madison 1 Linear Alg. Applies., vols. 162-4, 1992, 65-105 2Camhridge University Press, 1991.

The consecutive-k system was first studied around 1980, and it soon became a very popular subject. The reasons were many-folded, includ­ ing: 1. The system is simple and natural. So most people can understand it and many can do some analysis. Yet it can grow in many directions and there is no lack of new topics. 2. The system is simple enough to become a prototype for demonstrat­ ing various ideas related to reliability. For example, the interesting concept of component importance works best with the consecutive-k system. 3. The system is supported by many applications. Twenty years have gone and hundreds of papers have been published on the subject. This seems to be a good time for retrospect and to sort the scattered material into a book. Besides providing our own per­ spective, the book will also serve as an easy reference to the numerous ramifications of the subject. It is hoped that a summary of work in the current period will become the seed of future break-through.

The switching net.work is an important. classic research area in t.ele- communication and comput.er net.works. It.s import.ancc st.ems from both theory and practice. In fact, some open problems, such as Benes conjec- ture on shuffle-exchange networks and Chung-Rmis conjecture on multi- rate rearrangeability, still attract many researchers and the further de- velopment in optical networks requires advanced technology in optical switching networks. In 1997, we had a workshop in switching networks held in NSF Sci- ence and Technology Center in Discrete Mathematics and Theoretical Computer Science (DIMACS), at Princeton University. This workshop was very successful. Many participants wished to have a similar activity every two or three years. This book is a result of such a wish. We are putting together SOllle important developments in this area during last. several years, including articles ill fault-tolerance, rearrang{~ability. non- blocking, optical networks. random permutation generat.ioll. and layout complexity. SOlllC of thos(~ art ides are research papers alld SOIllC an' sur- veys. All articles were reviewed.We would like to uWlItioll two special problems studied in those articles.

Optical networks with terabits per second bandwith have received significant interest from both researchers and practitioners. This book captures a collection of research and survey papers presenting developments in this area. Contributions are from active researchers and cover a wide range of topics, including static and dynamic wavelength assignment algorithms, optimized wavelength converter allocation, traffic scheduling for QoS support, connection management, multicast routing, terabit packet switch architectures, multifibre networks, and multistage interconnection networks. The articles summarize the existing techniques, current developments and future directions as well as propose novel solutions to some important problems.

This work provides the most basic problems, concepts, and well-established results from the topological structure and analysis of interconnection networks in the graph-theoretic language. It covers the basic principles and methods of network design, several well-known networks such as hypercubes, de Bruijn digraphs, Kautz digraphs, double loop, and other networks, and the newest parameters to measure performance of fault-tolerant networks such as Menger number, Rabin number, fault-tolerant diameter, wide-diameter, restricted connectivity, and (l,w)-dominating number.

The exponential scaling of feature sizes in semiconductor technologies has side-effects on layout optimization, related to effects such as interconnect delay, noise, crosstalk, signal integrity, parasitics effects and power dissipation, that invalidate the assumptions that form the basis of previous design methodologies and tools. This book is intended to sample the most important, contemporary and advanced layout optimization problems emerging with the advent of very deep submicron technologies in semiconductor processing. A reference work for graduate students, senior undergraduates and researchers.

Along with the explosive growth of network traffic, multiwavelength optical networks are believed to be the technology that will support the ever-increasing bandwidth demands and will be the backbone for the next generation of the Internet. This title systematically studies the major research issues in WDM (Wavelength Division Multiplexing) optical networks, such as routing and wavelength assignment, QoS multicast routing, design of logical topologies, and placement of wavelength converters. The book consists of two parts. The first part studies the fundamental concepts and principles of WDM networks. The second part discusses advanced and research issues of WDM networks.

This title considers a graph as a mathematical structure on a set of elements with a binary relation. It covers basic concepts, trees and graphic spaces, plane graphs and planar graphs, flows and connectivity, matchings and independent sets, colouring theory, graphs and groups. These topics, both theoretical and applied, are treated with some depth and with some suggestions for further reading. The treatment of material particularly lays stress on digraphs, the mutual connections among these topics and the equivalence of some well-known theorems. All theorems are stated clearly, together with full and concise proofs. A number of examples, more than 350 figures and more than 500 exercises are given to help the reader understand and examine the materials covered in the book.

This text presents a specialized topic in the area of database management. Every researcher working in this area may use some of the techniques presented here. The complexity of the subject is handled in a detailed way so as to give down-to-earth explanations. One of the unique features of the work is the presentation of a collective view of common issues prevailing in several domains, including distributed databases. This approach enables the reader to appreciate the use of the tools and techniques developed, in the context of distributed databases applied to several domains, thus favoring multi-disciplinary research activity.

Clustering is an important technique for discovering relatively dense sub-regions or sub-spaces of a multi-dimension data distribution. Clus­ tering has been used in information retrieval for many different purposes, such as query expansion, document grouping, document indexing, and visualization of search results. In this book, we address issues of cluster­ ing algorithms, evaluation methodologies, applications, and architectures for information retrieval. The first two chapters discuss clustering algorithms. The chapter from Baeza-Yates et al. describes a clustering method for a general metric space which is a common model of data relevant to information retrieval. The chapter by Guha, Rastogi, and Shim presents a survey as well as detailed discussion of two clustering algorithms: CURE and ROCK for numeric data and categorical data respectively. Evaluation methodologies are addressed in the next two chapters. Ertoz et al. demonstrate the use of text retrieval benchmarks, such as TRECS, to evaluate clustering algorithms. He et al. provide objective measures of clustering quality in their chapter. Applications of clustering methods to information retrieval is ad­ dressed in the next four chapters. Chu et al. and Noel et al. explore feature selection using word stems, phrases, and link associations for document clustering and indexing. Wen et al. and Sung et al. discuss applications of clustering to user queries and data cleansing. Finally, we consider the problem of designing architectures for infor­ mation retrieval. Crichton, Hughes, and Kelly elaborate on the devel­ opment of a scientific data system architecture for information retrieval.

Since the outstanding and pioneering research work of Hopfield on recurrent neural networks (RNNs) in the early 80s of the last century, neural networks have rekindled strong interests in scientists and researchers. Recent years have recorded a remarkable advance in research and development work on RNNs, both in theoretical research as weIl as actual applications. The field of RNNs is now transforming into a complete and independent subject. From theory to application, from software to hardware, new and exciting results are emerging day after day, reflecting the keen interest RNNs have instilled in everyone, from researchers to practitioners. RNNs contain feedback connections among the neurons, a phenomenon which has led rather naturally to RNNs being regarded as dynamical systems. RNNs can be described by continuous time differential systems, discrete time systems, or functional differential systems, and more generally, in terms of non­ linear systems. Thus, RNNs have to their disposal, a huge set of mathematical tools relating to dynamical system theory which has tumed out to be very useful in enabling a rigorous analysis of RNNs.

Wireless networking enables two or more computers to communicate using standard network protocols without network cables. Since their emergence in the 1970s, wireless networks have become increasingly pop­ ular in the computing industry. In the past decade, wireless networks have enabled true mobility. There are currently two versions of mobile wireless networks. An infrastructure network contains a wired backbone with the last hop being wireless. The cellular phone system is an exam­ ple of an infrastructure network. A multihop ad hoc wireless network has no infrastructure and is thus entirely wireless. A wireless sensor network is an example of a multihop ad hoc wireless network. Ad hoc wireless networking is a technique to support robust and ef­ ficient operation in mobile wireless networks by incorporating routing functionality into mobile hosts. This technique will be used to realize the dream of "anywhere and anytime computing", which is termed mo­ bile computing. Mobile computing is a new paradigm of computing in which users carrying portable devices have access to shared infrastruc­ ture in any location at any time. Mobile computing is a very challenging topic for scientists in computer science and electrical engineering. The representative system for ad hoc wireless networking is called MANET, an acronym for "Mobile Ad hoc NETworks". MANET is an autonomous system consisting of mobile hosts connected by wireless links which can be quickly deployed.

This is the first book that provides readers with a deep technical overview of recent advances in resource management for wireless networks at different layers of the protocol stack. The subject is explored in various wireless networks, such as ad hoc wireless networks, 3G/4G cellular, IEEE 802.11, and Bluetooth personal area networks. Survey chapters give an excellent introduction to key topics in resource management for wireless networks, while experts will be satisfied by the technical depth of the knowledge imparted in chapters exploring hot research topics. The subject area discussed in this book is very relevant today, considering the recent remarkable growth of wireless networking and the convergence of wireless personal communications, internet technologies and real-time multimedia. This volume is a very good companion for practitioners working on implementing solutions for multimedia and Quality of Service -- sensitive applications over wireless networks.

Combinatorics and Matrix Theory have a symbiotic, or mutually beneficial, relationship. This relationship is discussed in my paper The symbiotic relationship of combinatorics and matrix theoryl where I attempted to justify this description. One could say that a more detailed justification was given in my book with H. J. Ryser entitled Combinatorial Matrix Theon? where an attempt was made to give a broad picture of the use of combinatorial ideas in matrix theory and the use of matrix theory in proving theorems which, at least on the surface, are combinatorial in nature. In the book by Liu and Lai, this picture is enlarged and expanded to include recent developments and contributions of Chinese mathematicians, many of which have not been readily available to those of us who are unfamiliar with Chinese journals. Necessarily, there is some overlap with the book Combinatorial Matrix Theory. Some of the additional topics include: spectra of graphs, eulerian graph problems, Shannon capacity, generalized inverses of Boolean matrices, matrix rearrangements, and matrix completions. A topic to which many Chinese mathematicians have made substantial contributions is the combinatorial analysis of powers of nonnegative matrices, and a large chapter is devoted to this topic. This book should be a valuable resource for mathematicians working in the area of combinatorial matrix theory. Richard A. Brualdi University of Wisconsin - Madison 1 Linear Alg. Applies., vols. 162-4, 1992, 65-105 2Camhridge University Press, 1991.

This book provides the most basic problems, concepts, and well-established results from the topological structure and analysis of interconnection networks in the graph-theoretic language. It covers the basic principles and methods of network design, several well-known networks such as hypercubes, de Bruijn digraphs, Kautz digraphs, double loop, and other networks, and the newest parameters to measure performance of fault-tolerant networks such as Menger number, Rabin number, fault-tolerant diameter, wide-diameter, restricted connectivity, and (l,w)-dominating number. Audience: The book is suitable for those readers who are working on or intend to start research in design analysis of the topological structure of interconnection networks, particularly undergraduates and postgraduates specializing in computer science and applied mathematics.

The exponential scaling of feature sizes in semiconductor technologies has side-effects on layout optimization, related to effects such as interconnect delay, noise, crosstalk, signal integrity, parasitics effects, and power dissipation, that invalidate the assumptions that form the basis of previous design methodologies and tools. This book is intended to sample the most important, contemporary, and advanced layout optimization problems emerging with the advent of very deep submicron technologies in semiconductor processing. Audience: A reference work for graduate students, senior undergraduates, and researchers.

Multiwavelength Optical Networks systematically studies the major research issues in WDM (Wavelength Division Multiplexing) optical networks, such as routing and wavelength assignment, QoS multicast routing, design of logical topologies, and placement of wavelength converters. The book consists of two parts. The first part studies the fundamental concepts and principles of WDM networks. The second part discusses advanced and research issues of WDM networks. The authors of the book have many years of working experience in the areas of computer networks and network optimization. The book discusses many difficult issues of WDM networks in a very comprehensive way. For each problem, there is a background discussion, and then the mathematical formulation, followed by the solutions.

In the spectrum of mathematics, graph theory which studies a mathe­ matical structure on a set of elements with a binary relation, as a recognized discipline, is a relative newcomer. In recent three decades the exciting and rapidly growing area of the subject abounds with new mathematical devel­ opments and significant applications to real-world problems. More and more colleges and universities have made it a required course for the senior or the beginning postgraduate students who are majoring in mathematics, computer science, electronics, scientific management and others. This book provides an introduction to graph theory for these students. The richness of theory and the wideness of applications make it impossi­ ble to include all topics in graph theory in a textbook for one semester. All materials presented in this book, however, I believe, are the most classical, fundamental, interesting and important. The method we deal with the mate­ rials is to particularly lay stress on digraphs, regarding undirected graphs as their special cases. My own experience from teaching out of the subject more than ten years at University of Science and Technology of China (USTC) shows that this treatment makes hardly the course di:fficult, but much more accords with the essence and the development trend of the subject.

N etwork-based computing domain unifies all best research efforts presented from single computer systems to networked systems to render overwhelming computational power for several modern day applications. Although this power is expected to grow with respect to time due to tech­ nological advancements, application requirements impose a continuous thrust on network utilization and on the resources to deliver supreme quality of service. Strictly speaking, network-based computing dornain has no confined scope and each element offers considerable challenges. Any modern day networked application strongly thrives on efficient data storage and management system, which is essentially a Database System. There have been nurnber of books-to-date in this domain that discuss fundamental principles of designing a database systern. Research in this dornain is now far matured and rnany researchers are venturing in this dornain continuously due to a wide variety of challenges posed. In this book, our dornain of interest is in exposing the underlying key challenges in designing algorithms to handle unpredictable requests that arrive at a Distributed Database System(DDBS) and evaluating their performance. These requests are otherwise called as on-line requests arriving at a system to process. Transactions in an on-line Banking service, Airline Reservation systern, Video-on-Demand systern, etc, are few examples of on-line requests.

These days, the nature of services and the volume of demand in the telecommu­ nication industry is changing radically, with the replacement of analog transmis­ sion and traditional copper cables by digital technology and fiber optic transmis­ sion equipment. Moreover, we see an increasing competition among providers of telecommunication services, and the development of a broad range of new services for users, combining voice, data, graphics and video. Telecommunication network planning has thus become an important problem area for developing and applying optimization models. Telephone companies have initiated extensive modeling and planning efforts to expand and upgrade their transmission facilities, which are, for most national telecommunication networks, divided in three main levels (see Balakrishnan et al. [5]), namely, l. the long-distance or backbone network that typically connects city pairs through gateway nodes; 2. the inter-office or switching center network within each city, that interconnects switching centers in different subdivisions (clusters of customers) and provides access to the gateway(s) node(s); 1 2 DESIGN OF SURVNABLE NETWORKS WITH BOUNDED RINGS 3. the local access network that connects individual subscribers belonging to a cluster to the corresponding switching center. These three levels differ in several ways including their design criteria. Ideally, the design of a telecommunication network should simultaneously account for these three levels. However, to simplify the planning task, the overall planning problem is decomposed by considering each level separately.

The huge bandwidth of optical fiber was recognized back in the 1970s during the early development of fiber optic technology. For the last two decades, the capacity of experimental and deployed systems has been increasing at a rate of 100-fold each decade-a rate exceeding the increase of integrated circuit speeds. Today, optical communication in the public communication networks has developed from the status of a curiosity into being the dominant technology. Various great challenges arising from the deployment of the wavelength division multiplexing (WDM) have attracted a lot of efforts from many researchers. Indeed, the optical networking has been a fertile ground for both theoretical researches and experimental studies. This monograph presents the contribution from my past and ongoing research in the optical networking area. The works presented in this book focus more on graph-theoretical and algorithmic aspects of optical networks. Although this book is limited to the works by myself and my coauthors, there are many outstanding achievements made by other individuals, which will be cited in many places in this book. Without the inspiration from their efforts, this book would have never been possible. This monograph is divided into four parts: • Multichannel Optical Networking Architectures, • Broadcast-and-Select Passive Optical Networks, • Wavelength-Switched Optical Networks, • SONET/WDM Optical Networks. The first part consists of the first three chapters. Chapter 1 pro­ vides a brief survey on the networking architectures of optical trans- XVll xvm MULTICHANNEL OPTICAL NETWORKS port networks, optical access networks and optical premise networks.

Clustering is an important technique for discovering relatively dense sub-regions or sub-spaces of a multi-dimension data distribution. Clus­ tering has been used in information retrieval for many different purposes, such as query expansion, document grouping, document indexing, and visualization of search results. In this book, we address issues of cluster­ ing algorithms, evaluation methodologies, applications, and architectures for information retrieval. The first two chapters discuss clustering algorithms. The chapter from Baeza-Yates et al. describes a clustering method for a general metric space which is a common model of data relevant to information retrieval. The chapter by Guha, Rastogi, and Shim presents a survey as well as detailed discussion of two clustering algorithms: CURE and ROCK for numeric data and categorical data respectively. Evaluation methodologies are addressed in the next two chapters. Ertoz et al. demonstrate the use of text retrieval benchmarks, such as TRECS, to evaluate clustering algorithms. He et al. provide objective measures of clustering quality in their chapter. Applications of clustering methods to information retrieval is ad­ dressed in the next four chapters. Chu et al. and Noel et al. explore feature selection using word stems, phrases, and link associations for document clustering and indexing. Wen et al. and Sung et al. discuss applications of clustering to user queries and data cleansing. Finally, we consider the problem of designing architectures for infor­ mation retrieval. Crichton, Hughes, and Kelly elaborate on the devel­ opment of a scientific data system architecture for information retrieval.

Wireless networking enables two or more computers to communicate using standard network protocols without network cables. Since their emergence in the 1970s, wireless networks have become increasingly pop­ ular in the computing industry. In the past decade, wireless networks have enabled true mobility. There are currently two versions of mobile wireless networks. An infrastructure network contains a wired backbone with the last hop being wireless. The cellular phone system is an exam­ ple of an infrastructure network. A multihop ad hoc wireless network has no infrastructure and is thus entirely wireless. A wireless sensor network is an example of a multihop ad hoc wireless network. Ad hoc wireless networking is a technique to support robust and ef­ ficient operation in mobile wireless networks by incorporating routing functionality into mobile hosts. This technique will be used to realize the dream of "anywhere and anytime computing", which is termed mo­ bile computing. Mobile computing is a new paradigm of computing in which users carrying portable devices have access to shared infrastruc­ ture in any location at any time. Mobile computing is a very challenging topic for scientists in computer science and electrical engineering. The representative system for ad hoc wireless networking is called MANET, an acronym for "Mobile Ad hoc NETworks". MANET is an autonomous system consisting of mobile hosts connected by wireless links which can be quickly deployed.

The consecutive-k system was first studied around 1980, and it soon became a very popular subject. The reasons were many-folded, includ­ ing: 1. The system is simple and natural. So most people can understand it and many can do some analysis. Yet it can grow in many directions and there is no lack of new topics. 2. The system is simple enough to become a prototype for demonstrat­ ing various ideas related to reliability. For example, the interesting concept of component importance works best with the consecutive-k system. 3. The system is supported by many applications. Twenty years have gone and hundreds of papers have been published on the subject. This seems to be a good time for retrospect and to sort the scattered material into a book. Besides providing our own per­ spective, the book will also serve as an easy reference to the numerous ramifications of the subject. It is hoped that a summary of work in the current period will become the seed of future break-through.

The switching net.work is an important. classic research area in t.ele- communication and comput.er net.works. It.s import.ancc st.ems from both theory and practice. In fact, some open problems, such as Benes conjec- ture on shuffle-exchange networks and Chung-Rmis conjecture on multi- rate rearrangeability, still attract many researchers and the further de- velopment in optical networks requires advanced technology in optical switching networks. In 1997, we had a workshop in switching networks held in NSF Sci- ence and Technology Center in Discrete Mathematics and Theoretical Computer Science (DIMACS), at Princeton University. This workshop was very successful. Many participants wished to have a similar activity every two or three years. This book is a result of such a wish. We are putting together SOllle important developments in this area during last. several years, including articles ill fault-tolerance, rearrang{~ability. non- blocking, optical networks. random permutation generat.ioll. and layout complexity. SOlllC of thos(~ art ides are research papers alld SOIllC an' sur- veys. All articles were reviewed.We would like to uWlItioll two special problems studied in those articles.