Gap+Theorem+on+Complete+Noncompact+Riemannian+Manifold
人工智能AI革命外文翻译中英文
人工智能（AI）革命外文翻译中英文英文The forthcoming Artificial Intelligence (AI) revolution:Its impact on society and firmsSpyros MakridakisAbstractThe impact of the industrial and digital (information) revolutions has, undoubtedly, been substantial on practically all aspects of our society, life, firms and employment. Will the forthcoming AI revolution produce similar, far-reaching effects? By examining analogous inventions of the industrial, digital and AI revolutions, this article claims that the latter is on target and that it would bring extensive changes that will also affect all aspects of our society and life. In addition, its impact on firms and employment will be considerable, resulting in richly interconnected organizations with decision making based on th e analysis and exploitation of “big” data and intensified, global competition among firms. People will be capable of buying goods and obtaining services from anywhere in the world using the Internet, and exploiting the unlimited, additional benefits that will open through the widespread usage of AI inventions. The paper concludes that significant competitive advantages will continue to accrue to those utilizing the Internet widely and willing to take entrepreneurial risks in order to turn innovative products/services into worldwide commercial success stories. The greatest challenge facing societies and firms would be utilizing the benefits of availing AI technologies, providing vast opportunities for both new products/services and immense productivity improvements while avoiding the dangers and disadvantages in terms of increased unemployment and greater wealth inequalities.Keywords：Artificial Intelligence (AI)，Industrial revolution，Digital revolution，AI revolution，Impact of AI revolution，Benefits and dangers of AI technologies The rise of powerful AI will be either the best or the worst thing ever to happento humanity. We do not yet know which.Stephen HawkingOver the past decade, numerous predictions have been made about the forthcoming Artificial Intelligence (AI) Revolution and its impact on all aspects of our society, firms and life in general. This paper considers such predictions and compares them to those of the industrial and digital ones. A similar paper was written by this author and published in this journal in 1995, envisioning the forthcoming changes being brought by the digital (information) revolution, developing steadily at that time, and predicting its impact for the year 2015 (Makridakis, 1995). The current paper evaluates these 1995 predictions and their impact identifying hits and misses with the purpose of focusing on the new ones being brought by the AI revolution. It must be emphasized that the stakes of correctly predicting the impact of the AI revolution arefar reaching as intelligent machines may become our “final invention” that may end human supremacy (Barrat, 2013). There is little doubt that AI holds enormous potential as computers and robots will probably achieve, or come close to, human intelligence over the next twenty years becoming a serious competitor to all the jobs currently performed by humans and for the first time raising doubt over the end of human supremacy.This paper is organized into four parts. It first overviews the predictions made in the 1995 paper for the year 2015, identifying successes and failures and concluding that major technological developments (notably the Internet and smartphones) were undervalued while the general trend leading up to them was predicted correctly. Second, it investigates existing and forthcoming technological advances in the field of AI and the ability of computers/machines to acquire real intelligence. Moreover, it summarizes prevailing, major views of how AI may revolutionize practically everything and its impact on the future of humanity. The third section sums up the impact of the AI revolution and describes the four major scenarios being advocated, as well as what could be done to avoid the possible negative consequences of AI technologies. The fourth section discusses how firms will be affected by these technologies that will transform the competitive landscape, how start-up firms are founded and the way success can be achieved. Finally, there is a brief concluding section speculating about the future of AI and its impact on our society, life, firms and employment.1. The 1995 paper: hits and missesThe 1995 paper (Makridakis, 1995) was written at a time when the digital (at that time it was called information) revolution was progressing at a steady rate. The paper predicted that by 2015 “the information revolution should be in full swing” and that “computers/communications” would be in widespread use, whi ch has actually happened, although its two most important inventions (the Internet and smartphones) and their significant influence were not foreseen as such. Moreover, the paper predicted that “a single computer (but not a smartphone) can, in addition to its traditional tasks, also become a terminal capable of being used interactively for the following:” (p. 804–805)• Picture phone and teleconference• Television and videos• Music• Shopping• On line banking and financial services• Reservations• Medic al advice• Access to all types of services• Video games• Other games (e.g., gambling, chess etc.)• News, sports and weather reports• Access to data banksThe above have all materialized and can indeed be accessed by computer,although the extent of their utilization was underestimated as smartphones are now being used widely. For instance, the ease of accessing and downloading scientific articles on one's computer in his/her office or home would have seemed like science fiction back in 1995, when finding such articles required spending many hours in the library (often in its basement for older publications) and making photocopies to keep them for later use. Moreover, having access, from one's smartphone or tablet, to news from anywhere in the world, being able to subscribe to digital services, obtain weather forecasts, purchase games, watch movies, make payments using smartphones and a plethora of other, useful applications was greatly underestimated, while the extensive use of the cloud for storing large amounts of data for free was not predicted at all at that time. Even in 1995 when the implications of Moore's law leading to increasing computer speed and storage while reducing costs were well known, nevertheless, it was hard to imagine that in 2016 there would be 60 trillion web pages, 2.5 billion smartphones, more than 2 billion personal computers and 3.5 billion Google searches a day.The paper correctly predicted “as wireless telecommunications will be possible the above list of capabilities can be accessed from anywhere in the world without the need for regular telephone lines”. What the 1995 paper missed, however, was that in 2015 top smartphones, costing less than €500, would be as powerful as the 1995 supercomputer, allowing access to the Internet and all tasks that were only performed by expensive computers at that time, including an almost unlimited availability of new, powerful apps providing a large array of innovative services that were not imagined twenty years ago. Furthermore, the paper correctly predicted super automation leading to unattended factories stating that “by 2015 there will be little need for people to do repetitive manual or mental tasks”. It also foresaw the decline of large industrial firms, increased global competition and the drop in the percentage of labour force employed in agriculture and manufacturing (more on these predictions in the section The Impact of the AI Revolution on Firms). It missed however the widespread utilization of the Internet (at that time it was a text only service), as well as search engines (notably Google), social networking sites(notably Facebook) and the fundamental changes being brought by the widespread use of Apple's iPhone, Samsung's Galaxy and Google's Android smartphones. It is indeed surprising today to see groups of people in a coffee shop or restaurant using their smartphones instead of speaking to each other and young children as little as three or four years of age playing with phones and tablets. Smartphones and tablets connected to the Internet through Wi-Fi have influenced social interactions to a significant extent, as well as the way we search for information, use maps and GPS for finding locations, and make payments. These technologies were not predicted in the 1995 paper.2. Towards the AI revolutionThe 1995 paper referred to Say, the famous French economist, who wrote in 1828 about the possibility of cars as substitutes for horses:“Nevertheless no machine will ever be able to perform what even the worst horses can - the service of carrying people and goods through the bustle and throng of a great city.” (p. 800)Say could never have dreamed of, in his wildest imagination, self-driving cars, pilotless airplanes, Skype calls, super computers, smartphones or intelligent robots. Technologies that seemed like pure science fiction less than 190 years ago are available today and some like self-driving vehicles will in all likelihood be in widespread use within the next twenty years. The challenge is to realistically predict forthcoming AI technologies without falling into the same short-sighted trap of Say and others, including my 1995 paper, unable to realize the momentous, non-linear advancements of new technologies. There are two observations to be made.First, 190 years is a brief period by historical standards and during this period we went from horses being the major source of transportation to self-driving cars and from the abacus and slide rules to powerful computers in our pockets. Secondly, the length of time between technological inventions and their practical, widespread use is constantly being reduced. For instance, it took more than 200 years from the time Newcomen developed the first workable steam engine in 1707 to when Henry Ford built a reliable and affordable car in 1908. It took more than 90 years between the time electricity was introduced and its extensive use by firms to substantially improve factory productivity. It took twenty years, however, between ENIAC, the first computer, and IBM's 360 system that was mass produced and was affordable by smaller business firms while it took only ten years between 1973 when Dr Martin Cooper made the first mobile call from a handheld device and its public launch by Motorola. The biggest and most rapid progress, however, took place with smartphones which first appeared in 2002 and saw a stellar growth with the release of new versions possessing substantial improvements every one or two years by the likes of Apple, Samsung and several Chinese firms. Smartphones, in addition to their technical features, now incorporate artificial intelligence characteristics that include understanding speech, providing customized advice in spoken language, completing words when writing a text and several other functions requiring embedded AI, provided by a pocket computer smaller in size than a pack of cigarettes.From smart machines to clever computers and to Artificial Intelligence (AI) programs: A thermostat is a simple mechanical device exhibiting some primitive but extremely valuable type of intelligence by keeping temperatures constant at some desired, pre-set level. Computers are also clever as they can be instructed to make extremely complicated decisions taking into account a large number of factors and selection criteria, but like thermostats such decisions are pre-programmed and based on logic, if-then rules and decision trees that produce the exact same results, as long as the input instructions are alike. The major advantage of computers is their lightning speed that allows them to perform billions of instructions per second. AI, on the other hand, goes a step further by not simply applying pre-programmed decisions, but instead exhibiting some learning capabilities.The story of the Watson computer beating Jeopardy's two most successful contestants is more complicated, since retrieving the most appropriate answer out of the 200 million pages of information stored in its memory is not a sign of real intelligence as it relied on its lightning speed to retrieve information in seconds. What is more challenging according to Jennings, one of Jeopardy's previous champions, is“to read clues in a natural language, understand puns and the red herrings, to unpack just the meaning of the clue” (May, 2013). Similarly, it is a sign of intelligence to improve it s performance by “playing 100 games against past winners”. (Best, 2016). Watson went several steps beyond Deep Blue towards AI by being able to understand spoken English and learn from his mistakes (New Yorker, 2016). However, he was still short of AlphaGo that defeated Go Champions in a game that cannot be won simply by using “brute force” as the number of moves in this game is infinite, requiring the program to use learning algorithms that can improve its performance as it plays more and more gamesComputers and real learning: According to its proponents, “the main focus of AI research is in teaching computers to think for themselves and improvise solutions to common problems” (Clark, 2015). But many doubt that computers can learn to think for themselves even though they can display signs of intelligence. David Silver, an AI scientist working at DeepMind, explained that “even though AlphaGo has affectively rediscovered the most subtle concepts of Go, its knowledge is implicit. The computer parse out these concepts –they simply emerge from its statistical comparisons of types of winning board positions at GO” (Chouard, 2016). At the same time Cho Hyeyeon, one of the strongest Go players in Korea commented that “AlphaGo seems like it knows everything!” while others believe that “AlphaGo is likely to start a ‘new revolution’ in the way we play Go”as “it is seeking simply to maximize its probability of reaching winning positions, rather than as human players tend to do –maximize territorial gains” (Chouard, 2016). Does it matter, as Silver said, that AlphaGo's knowledge of the game is implicit as long as it can beat the best players? A more serious issue is whether or not AlphaGo's ability to win games with fixed rules can extend to real life settings where not only the rules are not fixed, but they can change with time, or from one situation to another.From digital computers to AI tools: The Intel Pentium microprocessor, introduced in 1993, incorporated graphics and music capabilities and opened computers up to a large number of affordable applications extending beyond just data processing. Such technologies signalled the beginning of a new era that now includes intelligent personal assistants understanding and answering natural languages, robots able to see and perform an array of intelligent functions, self-driving vehicles and a host of other capabilities which were until then an exclusive human ability. The tech optimists ascertain that in less than 25 years computers went from just manipulating 0 and 1 digits, to utilizing sophisticated neural networkalgorithms that enable vision and the understanding and speaking of natural languages among others. Technology optimists therefore maintain there is little doubt that in the next twenty years, accelerated AI technological progress will lead to a breakthrough, based on deep learning that imitates the way young children learn, rather than the laborious instructions by tailor-made programs aimed for specific applications and based on logic, if-then rules and decision trees (Parloff, 2016).For instance, DeepMind is based on a neural program utilizing deep learning that teaches itself how to play dozens of Atari games, such as Breakout, as well or better than humans, without specific instructions for doing so, but by playing thousands ofgames and improving itself each time. This program, trained in a different way, became the AlphaGo that defeated GO champion Lee Sodol in 2016. Moreover, it will form the core of a new project to learn to play Starcraft, a complicated game based on both long term strategy as well as quick tactical decisions to stay ahead of an opponent, which DeepMind plans to be its next target for advancing deep learning (Kahn, 2016). Deep learning is an area that seems to be at the forefront of research and funding efforts to improve AI, as its successes have sparked a burst of activity in equity funding that reached an all-time high of more than $1 billion with 121 projects for start-ups in the second quarter of 2016, compared to 21 in the equivalent quarter of 2011 (Parloff, 2016).Google had two deep learning projects underway in 2012. Today it is pursuing more than 1000, according to their spokesperson, in all its major product sectors, including search, Android, Gmail, translation, maps, YouTube, and self-driving cars (The Week, 2016). IBM's Watson system used AI, but not deep learning, when it beat the two Jeopardy champions in 2011. Now though, almost all of Watson's 30 component services have been augmented by deep learning. Venture capitalists, who did not even know what deep learning was five years ago, today are wary of start-ups that do not incorporate it into their programs. We are now living in an age when it has become mandatory for people building sophisticated software applications to avoid click through menus by incorporating natural-language processing tapping deep learning (Parloff, 2016).How far can deep learning go? There are no limits according to technology optimists for three reasons. First as progress is available to practically everyone to utilize through Open Source software, researchers will concentrate their efforts on new, more powerful algorithms leading to cumulative learning. Secondly, deep learning algorithms will be capable of remembering what they have learned and apply it in similar, but different situations (Kirkpatrick et al., 2017). Lastly and equally important, in the future intelligent computer programs will be capable of writing new programs themselves, initially perhaps not so sophisticated ones, but improving with time as learning will be incorporated to be part of their abilities. Kurzweil (2005) sees nonbiological intelligence to match the range and subtlety of human intelligence within a quarter of a century and what he calls “Singularity” to occur by 2045, b ringing “the dawning of a new civilization that will enable us to transcend our biological limitations and amplify our creativity. In this new world, there will be no clear distinction between human and machine, real reality and virtual reality”.For some people these predictions are startling, with far-reaching implications should they come true. In the next section, four scenarios associated with the AI revolution are presented and their impact on our societies, life work and firms is discussed.3. The four AI scenariosUntil rather recently, famines, wars and pandemics were common, affecting sizable segments of the population, causing misery and devastation as well as a large number of deaths. The industrial revolution considerably increased the standards of living while the digital one maintained such rise and also shifted employment patterns,resulting in more interesting and comfortable office jobs. The AI revolution is promising even greater improvements in productivity and further expansion in wealth. Today more and more people, at least in developed countries, die from overeating rather than famine, commit suicide instead of being killed by soldiers, terrorists and criminals combined and die from old age rather than infectious disease (Harari, 2016). Table 1 shows the power of each revolution with the industrial one aiming at routine manual tasks, the digital doing so to routine mental ones and AI aiming at substituting, supplementing and/or amplifying practically all tasks performed by humans. The cri tical question is: “what will the role of humans be at a time when computers and robots could perform as well or better andmuch cheaper, practically all tasks that humans do at present?” There are four scenarios attempting to answer this question.The Optimists: Kurzweil and other optimists predict a “science fiction”, utopian future with Genetics, Nanotechnology and Robotics (GNR) revolutionizing everything, allowing humans to harness the speed, memory capacities and knowledge sharing ability of computers and our brain being directly connected to the cloud. Genetics would enable changing our genes to avoid disease and slow down, or even reverse ageing, thus extending our life span considerably and perhaps eventually achieving immortality. Nanotechnology, using 3D printers, would enable us to create virtually any physical product from information and inexpensive materials bringing an unlimited creation of wealth. Finally, robots would be doing all the actual work, leaving humans with the choice of spending their time performing activities of their choice and working, when they want, at jobs that interest them.The Pessimists: In a much quoted article from Wired magazine in 2000, Bill Joy (Joy, 2000) wrote “Our most powerful 21st-century technologies –robotics, genetic engineering, and nanotech –are threatening to make humans an endangered species”. Joy pointed out that as machines become more and more intelligent and as societal problems become more and more complex, people will let machines make all the important decisions for them as these decisions will bring better results than those made by humans. This situation will, eventually, result in machines being in effective control of all important decisions with people dependent on them and afraid to make their own choices. Joy and many other scientists (Cellan-Jones, 2014) and philosophers (Bostrom, 2014) believe that Kurzweil and his supporters vastly underestimate the magnitude of the challenge and the potential dangers which can arise from thinking machines and intelligent robots. They point out that in the utopian world of abundance, where all work will be done by machines and robots, humans may be reduced to second rate status (some saying the equivalent of computer pets) as smarter than them computers and robots will be available in large numbers and people will not be motivated to work, leaving computers/robots to be in charge of making all important decisions. It may not be a bad world, but it will definitely be a different one with people delegated to second rate status.Harari is the newest arrival to the ranks of pessimists. His recent book (Harari, 2016, p. 397) concludes with the following three statements:• “Science is converging to an all-encompassing dogma, which says thatorganisms are algorithm s, and life is data processing”• “Intelligence is decoupling from consciousness”• “Non-conscious but highly intelligent algorithms may soon know us better than we know ourselves”Consequently, he asks three key questions (which are actually answered by the above three statements) with terrifying implications for the future of humanity: • “Are organisms really just algorithms, and is life just data processing?”• “What is more valuable –intelligence or consciousness?”• “What will happen to society, polit ics and daily life when non-conscious but highly intelligent algorithms know us better than we know ourselves?”Harari admits that nobody really knows how technology will evolve or what its impact will be. Instead he discusses the implications of each of his three questions: • If indeed organisms are algorithms then thinking machines utilizing more efficient ones than those by humans will have an advantage. Moreover, if life is just data processing then there is no way to compete with computers that can consult/exploit practically all available information to base their decisions.• The non-conscious algorithms Google search is based on the consultation of millions of possible entries and often surprise us by their correct recommendations. The implications that similar, more advanced algorithms than those utilized by Google search will be developed (bearing in mind Google search is less than twenty years old) in the future and be able to access all available information from complete data bases are far reachi ng and will “provide us with better information than we could expect to find ourselves”.• Humans are proud of their consciousness, but does it matter that self-driving vehicles do not have one, but still make better decisions than human drivers, as can be confirmed by their significantly lower number of traffic accidents?When AI technologies are further advanced and self-driving vehicles are in widespread use, there may come a time that legislation may be passed forbidding or restricting human driving, even though that may still be some time away according to some scientists (Gomes, 2014). Clearly, self-driving vehicles do not exceed speed limits, do not drive under the influence of alcohol or drugs, do not get tired, do not get distracted by talking on the phone or sending SMS or emails and in general make fewer mistakes than human drivers, causing fewer accidents. There are two implications if humans are not allowed to drive. First, there will be a huge labour displacement for the 3.5 million unionized truck drivers in the USA and the 600 thousand ones in the UK (plus the additional number of non-unionized ones) as well as the more than one million taxi and Uber drivers in these two countries. Second, and more importantly, it will take away our freedom of driving, admitting that computers are superior to us. Once such an admission is accepted there will be no limits to letting computers also make a great number of other decisions, like being in charge of nuclear plants, setting public policies or deciding on optimal economic strategies as their biggest advantage is their objectivity and their ability to make fewer mistakes than humans.One can go as far as suggesting letting computers choose Presidents/PrimeMinisters and elected officials using objective criteria rather than having people voting emotionally and believing the unrealistic promises that candidates make. Although such a suggestion will never be accepted, at least not in the near future, it has its merits since people often choose the wrong candidate and later regret their choice after finding out that pre-election promises were not only broken, but they were even reversed. Critics say if computers do eventually become in charge of making all important decisions there will be little left for people to do as they will be demoted to simply observing the decisions made by computers, the same way as being a passenger in a car driven by a computer, not allowed to take control out of the fear of causing an accident. As mentioned before, this could lead to humans eventually becoming computers’ pets.The pragmatists: At present the vast majority of views about the future implications of AI are negative, concerned with its potential dystopian consequences (Elon Musk, the CEO of Tesla, says it is like “summoning the demon” and calls the consequences worse than what nuclear weapons can do). There are fewer optimists and only a couple of pragmatists like Sam Altman and Michio Kaku (Peckham, 2016) who believe that AI technologies can be controlled through “OpenAI” and effective regulation. The ranks of pragmatists also includes John Markoff (Markoff, 2016) who pointed out that the AI field can be distinguished by two categories: The first trying to duplicate human intelligence and the second to augment it by expanding human abilities exploiting the power of computers in order to augment human decision making. Pragmatists mention chess playing where the present world champion is neither a human nor a computer but rather humans using laptop computers (Baraniuk, 2015). Their view is that we could learn to exploit the power of computers to augment our own skills and always stay a step ahead of AI, or at least not be at a disadvantage. The pragmatists also believe that in the worst of cases a chip can be placed in all thinking machines/robots to render them inoperative in case of any danger. By concentrating research efforts on intelligence augmentation, they claim we can avoid or minimize the possible danger of AI while providing the means to stay ahead in the race against thinking machines and smart robots.The doubters: The doubters do not believe that AI is possible and that it will ever become a threat to humanity. Dreyfus (1972), its major proponent, argues that human intelligence and expertise cannot be replicated and captured in formal rules. He believes that AI is a fad promoted by the computer industry. He points out to the many predictions that did not materialize such as those made by Herbert A. Simon in 1958 that “a computer would be the world's chess champion within ten years” and those made in 1965 that “machines will be capable within twenty years, of doing any work a man can do” (Crevier, 1993). Dreyfus claims that Simon's optimism was totally unwarranted as they were based on false assumptions that human intelligence is based on an information processing viewpoint as our mind is nothing like a computer. Although, the doubters’ criticisms may have been valid in the last century, they cannot stand for the new developments in AI. Deep Blue became the world's chess champion in 1997 (missing Simon's forecast by twenty one years) while we are not far today from machines being capable of doing all the work that humans can do (missing。
Analysis and design of associative memories based on stability of cellular neural networks
This article appeared in a journal published by Elsevier.The attached copy is furnished to the author for internal non-commercial research and education use,including for instruction at the authors institutionand sharing with colleagues.Other uses,including reproduction and distribution,or selling or licensing copies,or posting to personal,institutional or third partywebsites are prohibited.In most cases authors are permitted to post their version of thearticle(e.g.in Word or Tex form)to their personal website orinstitutional repository.Authors requiring further informationregarding Elsevier’s archiving and manuscript policies areencouraged to visit:/copyrightAnalysis and design of associative memories based on stability of cellular neural networksQi Han a,n,Xiaofeng Liao b,Tingwen Huang c,Jun Peng a,Chuandong Li b,Hongyu Huang ba School of Electrical and Information Engineering,Chongqing University of Science and Technology,Chongqing401331,Chinab State Key Laboratory of Power Transmission Equipment and System Security,College of Computer Science,Chongqing University,Chongqing400030,Chinac Texas A&M University at Qatar,Doha,P.O.Box23874,Qatara r t i c l e i n f oArticle history:Received6September2011Received in revised form14June2012Accepted17June2012Communicated by J.LiangAvailable online1July2012Keywords:Cellular neural networksAssociative memoriesCloning templatea b s t r a c tIt is well known that the prototype patterns in associative memories can be represented by stableequilibrium points of cellular neural networks(CNNs).Therefore,the stability of equilibrium points ofCNNs is critical in associative memories based on CNNs.In this paper,some criteria about the stabilityof CNNs are established.In fact,these criteria give some constraint conditions for the relationship ofparameters of pared with the previous works,our results relax the conservatism of therelationship of parameters and extend the range of the values of parameters.Two design procedures onthe parameters of CNNs are given to achieve associative memories under our criteria.Finally,anexample is given to verify the theoretical results and design procedures.&2012Elsevier B.V.All rights reserved.1.IntroductionCellular neural networks(CNNs)werefirst introduced in1988[1,2].CNNs can be arranged in matrix and implemented by somesimple analog circuits called cell.Each cell in CNNs is onlyconnected to its neighboring cells.Therefore,CNNs are wellfitfor very large-scale integration implementations due to this localinterconnection property,and have found many applications in avariety of areas,such as image processing[3],pattern recognition[4]and associative memories[5].In this paper,we mainly discussthe application of CNNs in associative memories.At its simplest,an associative memory is a system which stores mappings fromspecific input patterns to specific output patterns.That is to say,asystem which‘‘associates’’two patterns is that when one of twopatterns is presented,the other can be reliably recalled.There aretwo kinds of associative memories:auto-associative memoriesand hetero-associative.In associative memory,the retrievedpattern can be different from the probe in content or format,however,in some other search approaches,these two patternshave to be same or similar,such as similarity search in[6].Since Liu and Michel[7]reported that CNNs are effective as anassociative memories medium,associative memories havereceived a great deal of interest.Next,we would introduceassociative memories according to the time sequence.Sparselyinterconnected neural networks for associative memories werepresented in[8],and sparse synthesis technique was applied tothe design of a class of CNNs.A design algorithm for CNNs withspace-invariant cloning template with applications to associativememories was presented in[9].A synthesis procedure forassociative memories using discrete-time CNNs(DTCNNs)withlearning and forgetting capabilities was presented in[10].Asynthesis procedure of CNNs for associative memories wasintroduced in[11],where the method assured the global asymp-totic stability of the equilibrium point.DTCNNs with a globallyasymptotically stable equilibrium point were designed to behaveas associative memories in[12].In the last ten years,associativememories were realized by local stability of equilibrium points ofCNNs.In[13,14],the number of memory patterns of CNNs whichwere locally exponentially stable was obtained,and the designprocedures of associative memories based on CNNs was given.Adesign method for synthesizing associative memories based ondiscrete-time recurrent neural networks was presented in[15].In[16],a new design procedure for synthesizing associative mem-ories based on CNNs with time delays characterized by input andoutput matrices was introduced.From the above introduction about associative memories,it iseasy to know that stability of CNNs plays an important role inassociative memories.The prototype patterns in associativememories can be represented by stable equilibrium points ofcellular neural networks.Therefore,it is important to study thestability of equilibrium points of CNNs.There have been abundantresearches about stability of CNNs.Some sufficient conditions forContents lists available at SciVerse ScienceDirectjournal homepage:/locate/neucomNeurocomputing0925-2312/$-see front matter&2012Elsevier B.V.All rights reserved./10.1016/j.neucom.2012.06.017n Corresponding author.E-mail address:yiding1981@(Q.Han).Neurocomputing97(2012)192–200CNNs to be stable were obtained by constructing Lyapunov Function [17–24],and these conditions generally made equili-brium point global asymptotically stable.However,some authors presented some conditions which made equilibrium points locally stable,and there generally were multiple equilibrium points [13,14,25–27].In previous papers,the conditions of stability of CNNs are conservative.For example,bias vectors were computed by one of all memory patterns in [7–9];the relations between cloning templates were stronger and closer in [13–16].Therefore,the aim of the paper is to relax conservative relationship among parameters of CNNs.In order to get our theories,we choose the initial states of CNNs are zero.Our theories expend the scope of the values of parameters of CNNs.When the inputs and outputs of a CNN are given,the values of parameters can be obtained by our methods.In fact,if we get appropriate values of parameters of CNNs,we can realize associative memories.Therefore,from the above theoretical analysis,we give design procedures of associa-tive memories based on CNNs.The remaining parts of this paper are organized as follows.In Section 2,a class of CNNs are given.In Section 3,the relationship among parameters of CNNs is given by a theorem and some corollaries.These theories give some methods about how to get the value of parameters A ,D and V of a CNN.In Section 4,two design procedures on associative memories and a flow chart about how to get parameters of a CNN are given.In Section 5,an example is given to verify the theoretical results and design procedures.Some conclusions are finally drawn in Section 6.2.PreliminariesConsider a class of cellular neural networks defined by thefollowing differential equation:_y ij ðt Þ¼Àc ij y ij ðt ÞþX k 2ði ,r Þk ¼k 1ði ,r ÞX l 2ðj ,r Þl ¼l 1ðj ,r Þa kl g i þk ,j þl ðy ðt ÞÞþZ ij Z ij ¼X k 2ði ,r Þk ¼k 1ði ,r ÞX l 2ðj ,r Þl ¼l 1ðj ,r Þd kl u kl þv ij 8>>>>>><>>>>>>:ð1Þwhere y ij ðt ÞA R denotes the states vector,c ij is a positive para-meter,r is positive integer denoting neighborhood radius,A ¼ða kl Þð2r þ1Þð2r þ1Þa 0is intra-neuron connection weight matrix,D ¼ðkl Þð2r þ1Þð2r þ1Þis input cloning template,u kl is the input,v ij is the bias,k 1ði ,r Þ¼max f 1Ài ,Àr g ,k 2ði ,r Þ¼min f N Ài ,r g ,l 1ðj ,r Þ¼min f 1Àj ,Àr g ,l 2ðj ,r Þ¼max M Àj ,r g Èand g ðU Þis the activa-tion function defined by g ðy Þ¼ð9y þ19À9y À19Þ=2g i þk ,j þl ðy ðt ÞÞis output of a cell of CNNs.Outputs of CNNs can be memory patterns of associative memories.The expressions of template A and D are in Appendix A .Let r ¼1and n ¼NM .If the system (1)has N rows and M columns,then it can be put in vector form as _x¼ÀCx þAf ðx ÞþDU þV ð2Þwhere x ¼ðx 1,x 2,...,x n ÞT ¼ðy 11,y 12,...,y 1M ,...,y NM ÞT ,coefficient matrices A and D are obtained through the templates and C ¼diag ðc 1...c n Þ,the input vector U ¼ðu 1,...,u n ÞT ,t V ¼ðv 1,...,v n ÞT and f ðx Þ¼ðg ðy 1Þ,...,g ðy n ÞÞT .The k th cell in Eq.(2)is denoted by O k (k ¼iN þj ,where 1r i r N ,1r j r M ,i denotes i th row and j denotes j th column of the CNN).The expressions of template A and D are in Appendix A .The matrices of A and D are very complex.They are not only sparse matrices,and the posi-tions of their elements in these two matrices have certain rules.Therefore,these two matrices cannot be solved by previousmethods,such as the method in [28].In order to get the values of these two matrices,we need to transform these two matrices in two vectors.The methods will be shown in Section 3.2.Let a ¼ða 1,a 2,...,a n ÞTA U n ¼x i A R n x i ¼1or x i ¼À1,i ¼1,2,È...,n g ,C ða Þ¼x A R n Èx i a i 41,i ¼1,2,...,n g .Then,for x A C ða Þ,the Eq.(2)can be rewritten as _x¼ÀCx þA a þDU þV ð3ÞIf b is an equilibrium point of (3),then we haveb ¼C À1ðA a þDU þV ÞA C ða Þð4ÞLemma 1.[7]Suppose a ¼ða 1,a 2,...,a n ÞT A U n .If b ¼ðb 1,b 2,...,b n ÞT ¼C À1ðA a þDU þV ÞA C ða Þ,then b is an asymptotically stable equilibrium point of (2).Proof.Eq.(3)has a unique equilibrium point at x e ¼C À1ðA a þDU þV Þ,and x e ¼b A C ða Þby assumption.Therefore,this equili-brium is also asymptotical stable,since Eq.(3)has all its n eigenvalues at Àc i ,i ¼1,2,...,n .3.Main resultIn this section,we will give some theories about stability of CNNs firstly.Then,some methods are obtained on the basis of these theories for realizing associative memories based on CNNs.3.1.Stability of CNNsFirst,we give a theorem and some corollaries for achieving associative memories.The Eq.(3)can be rewritten as _xi ¼Àc i x i þX n j ¼1a ij a j þX n j ¼1d ij u j þv i ,i ¼1,2,...,n ð5ÞTheorem 1.In Eq.(5),let x i ð0Þ¼0,i ¼1,2,...,n :(i).If Pn j ¼1a ij a j þP n j ¼1d ij u j þv i 4c i ,then the Eq.(5)convergesto a positive stable equilibrium point,and the value of positive equilibrium point is bigger than 1.(ii).If P n j ¼1a ij a j þP nj ¼1d ij u j þv i o Àc i ,then the Eq.(5)con-verges to a negative stable equilibrium point,and the valuesof negative equilibrium point is less than À1.The proof of Theorem 1see Appendix B1.By the above theorem,we can get the following corollaries immediately.Corollary 1.In Eq.(5),let x i ð0Þ¼0and a ii Z c i ,i ¼1,2,...,n ,(i)If P n j ¼1,j a i a ij a j þP nj ¼1d ij u j þv i 40,then the Eq.(5)con-verges to a positive stable equilibrium point,and the valueof positive equilibrium point is bigger than 1.(ii)If P n j ¼1,j a i a ij a j þP nj ¼1d ij u j þv i o 0,then the Eq.(5)con-verges to a negative stable equilibrium point,and the valueof negative equilibrium point is less than À1.The proof of Theorem 1see Appendix B2.Corollary 2.In Eq.(5),let x i ð0Þ¼0and a ii o c i ,(i)If P n j ¼1a ij a j þP nj ¼1d ij u j þv i 4c i ,then the Eq.(5)convergesto a positive stable equilibrium point,and the value of positive equilibrium point is bigger than 1.Q.Han et al./Neurocomputing 97(2012)192–200193(ii)If P n j ¼1a ij a j þP nj ¼1d ij u j þv i o Àc i ,then the Eq.(5)convergesto a negative stable equilibrium point,and the value of negative equilibrium point is less than À1.The proof of Corollary 2is same to that of Theorem 1.If we choose v i ¼0in Theorem 1,then we can get the following corollary:Corollary 3.In Eq.(5),let x i ð0Þ¼0and v i ¼0,(i)If P n j ¼1a ij a j þP nj ¼1d ij u j 4c i ,then the Eq.(5)converges to apositive equilibrium point,and the value of positive equili-brium point is bigger than 1.(ii)If P n j ¼1a ij a j þP nj ¼1d ij u j o Àc i ,then the Eq.(5)converges to anegative equilibrium point,and the value of negative equili-brium point is less than À1.If we choose v i ¼0in Corollary 1,then we can get the following corollary:Corollary 4.In Eq.(5),let x i ð0Þ¼0,v i ¼0and a ii Z c i ,(i).If P nj ¼1,j a i a ij a j þP nj ¼1d ij u j 40,then the Eq.(5)converges to a positive stable equilibrium point,and the value of positive equilibrium point is bigger than 1.(ii).If Pn j ¼1,j a i a ij a j þP n j ¼1d ij u j o 0,then the Eq.(5)converges toa negative stable equilibrium point,and the value of negative equilibrium point is less than À1.Remark 1.In Theorem 1and its corollaries,though the initial states of a CNN have to return to zero,our methods does not strictly limit the relations between cloning template of the CNN.For example,P n j ¼1ð9a ij þb ij 9Þo c i o 1and P 1i ¼À1P 1i ¼À1ð9a ij þij 9Þo 1in [15,16],however our theories do not exist these limitations.The Theorem 1and its corollaries will be used to realize associative memories based on CNNs in the following.In order to realize associative memories,inputs of the CNN and the memory patterns(outputs of the CNN)should be given.Then,according to the above theories,we train the CNN on the basis of inputs and outputs,and the parameters (weight value)of the CNN can be obtained.Actually,the above theorem and corollaries give some methods or constraints about how to get parameters of a CNN.For example,when initial states of a CNN are zero,the relationship among parameters can be obtained by Theorem 1.If the values of some parameters of a CNN are fixed,the regions of values of other parameters of the CNN can be obtained.If inputs,outputs and parameters A ,C and D are fixed,the region of bias V can be gotten.The main difference between Theorem 1and Corollary 3is whether all bias v i (i ¼1,2,y n )are equal to 0.Similarly,the main difference of Corollary 1and Corollary 4is also whether all bias v i (i ¼1,2,y n )are equal to 0.The main difference between Theorem 1and Corollary 1is that whether there must exists a ii Z c i .Similarly,the main difference of Corollary 3and Corollary 4is also that whether there must exists a ii Z c i .In Appendix C ,we give a lot of notations.These notations will be used in matrix equations of next section.3.2.Associative memoriesNext,we will discuss associative memories based on CNNs by use of the above the theories and notations in Appendix C .In the section,the explanation of new notations can be found in Appendix C .First,we discuss how to get cloning template A and D of a CNN.We can make use of the inputs and outputs of cells in set R to get A and D .Because of the all values of v i (i ¼1,2,y n )of cells in set R are equal to zero,choose that all v i of all cells in set R are equal tozero.Then,by Corollaries 3or 4,the relationship among Pn j ¼1a ij a j ,P nj ¼1d ij u j,c i (i ¼1,y n )can be established.Then,we use Theorem 1or Corollary 1to calculate the regions of v i (i ¼1,2,y n )of cells in set P or Q .Therefore,in order to get parameters A ,D ,C ,and V for a CNN,we divide the question into two cases on the basis of Corollary 3and Corollary 4,respectively.(i)Corollary 3is used to obtain the relationship among parameters A ,D and C .Furthermore,the values of A and D can be gotten.Let l i 4max 1r i r nc i f g .We choose ^F 0¼0:5^D ,ð9Þand^AG þ^DU ¼^LG ð10ÞIt is obvious that Eq.(9)and Eq.(10)satisfy Corollary 3.FromEq.(9),the sign of Pn j ¼1a ij a j in a cell O i and the sign of output ofthe cell are the same.From Eq.(10),sign of P n j ¼1a ij a j þP nj ¼1d ij u j in a cell O i of a CNN and the sign of output of the cell are thesame.By Corollary 3,we know that sign of Pn j ¼1a ij a j þP n j ¼1d ij u j in a cell of a CNN can determine the sign of output of the cell.(ii)Corollary 4is used to achieve associative memories.Let l i 40.We choose ^F 00¼0:5^D ,ð11Þand~AG þ^DU ¼^LG ð12ÞIt is obvious that Eq.(11)and Eq.(12)satisfy Corollary 4.From Eqs.(9)–(12),we give the methods about computing parameters A and D .However,we find that it is difficult to beobtain ^Aand ^D through Eqs.(9)–(12).For example,in Eq.(10),the values of ^AG ,U and ^LG are known,and the value of ^D is unknown.We notice that there is no method to get the value of ^D.So,Eqs.(9)–(12)are needed to transform.Eq.(9)can be transformed as ^OLA ¼0:5^D ,ð13ÞTherefore,LA ¼0:5pinv ð^O0Þ^D ,ð14Þwhere pinv ðU Þdenotes pseudo inverse of a matrix.Eq.(10)can be transformed as ^O00LA ¼0:5^D ð15ÞTherefore,LA ¼0:5pinv ð^O00Þ^D ð16ÞEq.(11)can be transformed as ^XLD ¼^D À^F 0ð17ÞTherefore,LD ¼pinv ð^XÞð^D À^O 0LA Þð18ÞEq.(12)can be transformed as ^XLD ¼^D À^F 00ð19ÞTherefore,LD ¼pinv ð^XÞð^D À^O 00LA Þð20ÞQ.Han et al./Neurocomputing 97(2012)192–200194After transforming Eqs.(9)–(12),from LA and LD ,it is easy to obtain parameters A and D .Remark 2.When matrices ^O0,^O 00or ^X are irreversible,the values of LA or LD are approximate values.Next,we discuss how to get bias v i for a cell in sets P and Q .In set R ,the value of v i of a cell are equal to zero.However,in sets P and Q ,the values of v i (O i A P ,Q )of a cell are not equal to zero.Therefore,we need to calculate the value.Parameters A and D can be obtained by Corollary 3or 4.The ranges of v i (i ¼1,y n )can be obtained by Theorem 1or Corollary 1.From Theorem 1,we knowthat when v i 4c i ÀP n j ¼1a ij a j ÀP nj ¼1d ij u j ,the Eq.(5)converges to a positive stable equilibrium point,which means that the output of cellO i is positive.Because the sign of outputs of all cells in set P ispositive,we can choose that ^v þ4max 1r i r nf c i ÀPn j ¼1a ij a j ÀP nj ¼1d ij u j g is the bias of all cells in set P .Similarly,^v Ào min 1r i r n Àc i ÀP n j ¼1a ij a j ÀP nj ¼1d ij u j Én is the bias of all cells in set Q .Then,the regions of ^vþand ^vÀcan be get by Theorem 1or Corollary 1.We can consider the regions of ^vþand ^v Àfrom two aspects as follows.(i)If Theorem 1is used to realize associative memories based on a CNN,we can get the following result:If a i ¼1in Eq.(5),we get v i ðl Þ4c i ÀP n j ¼1a ij a l jÀP nj ¼1d ij u l j ¼x 0i ðl Þ.If a i ¼À1in Eq.(5),we get v i ðl Þo c i ÀP n j ¼1a ij a l j ÀP nj ¼1d ij u lj ¼x 0i ðl Þ.Therefore,we choose ^vþZ max1r l r m ,O i A P9f x 0i ðl Þg 9ð21Þas bias of all cells in set P ,and ^vÀr Àmax 1r l r m ,O i A Q9f x 0i ðl Þg 9ð22Þas bias of all cells in set Q .(ii)If Corollary 1is used to realize associative memories based on a CNN,we can get the following results:If a i ¼1in Eq.(5),we get v i ðl Þ4ÀP n j ¼1a ij a l j ÀP n j ¼1d ij u lj ¼x 00i ðl Þ.If a i ¼À1in Eq.(5),we get v i ðl Þo ÀP n j ¼1a ij a lj ÀP nj ¼1d iju lj ¼x 00i ðl Þ.Therefore,we choose ^vþZ max1r l r m ,O i A P9f x 00i ðl Þg 9ð23Þas bias of all cells in set P ,and ^vÀr Àmax 1r l r m ,O i A Q9f x 00i ðl Þg 9ð24Þas bias of all cells in set Q .Remark 3.Cloning template D is computed by the cells in set R .Therefore,the values of D are not be affected by the cells in sets P and Q ,which can make the values of D more accurate than that in [15].Bias vector is computed by the cells in sets P and Q ,which make the outputs of cells in sets P and Q always right in the process of associative memories.Remark 4.In [15],the authors proofed that associative memories based on CNNs are high-capacity (High-capacity means that a large number of memory patterns can be stored by a CNN).Themodel of the paper is similar with that of [15],therefore,we donot proof the problems of high-capacity.4.Design procedure of a CNNIn this section,two design procedures of parameters of a CNN are given by the above theories.(i)If Theorem 1and Corollary 3are chosen to achieve associativememories,we give the following design procedure of para-meters of a CNN.We can use Corollary 3to get parameters A and D in set R and use Theorem 1to get biases v i (i ¼1,y n )for a cell in sets P and Q .Step 1.Denote a matrix G ¼ða 1,a 2,...,a m Þas memory pat-terns of associative memories,where a i is a set of outputs of all cells in a CNN,and m is the number of patterns of associative memories.Denote a input matrixU ¼ðU 1,U 2,...,U mÞwhich is corresponding to the matrix of memory patterns G .Step 2.Divide all cells of the CNN into three sets.If all outputs of a cell in all memory patterns are 1,the cell will be classified as set P .If all outputs of a cell in all memory patterns are À1,the cell will be classified as set Q .If the outputs of a cell in all memory patterns are 1and À1,the cell will be classified as set R .Step 3.Choose biases v i ðO i A R Þare equal to zero.Step 4.Determine all state constants c ij ,1r i r N ,1r j r M .Obtain coefficient matrices C .Step 5.Determine matrix L such that l i 4max 1r i r nf c ig ,and getmatrix L 0.Step pute cloning template A by use of Eq.(14)and the relationship between inputs and outputs of cells in set R .Obtain coefficient matrices A .Step pute cloning template by use of Eq.(18)and the relationship between inputs and outputs of cells in set R .Obtain coefficient matrices D .Step pute x 0i ðl Þ(O i A P ,1r l r m )in set P .Choose ^vþZ max 1r l r m ,O i A P9f x 0i ðl Þg 9in terms of (21),and biases v i ðO i A P Þareequal to ^vþ.Step pute x 0i ðl Þ(O i A Q ,1r l r m )in set Q .Choose^vÀo Àmax 1r l r m ,O i A Q9f x 0i ðl Þg 9in terms of (22),and biases v i ðO i A Q Þare equal to ^vÀ.Step 10.Synthesize the CNN with the connection weight matrices A ,C ,D and bias vector V .From these above ten steps,we give the flow chart as in Fig.1:(ii)If Corollary 1and Corollary 4are chosen to realize associativememories,we give the following design procedure for a CNN:Step 1,Step 2,Step 3and Step 4of (ii)are same to these of (i).Step 5.Determine matrix L such that l i 40,and get matrix L 0.Step 6.Determine a 00such that a 00Z max 1r i r nf c ig holds.Step pute cloning template A from (16)in set R .Obtain coefficient matrices A .Step pute cloning template from (20)in set R .Obtain coefficient matrices D .Step pute x 00i ðl Þ(O i A P ,1r l r m )in set P .Choose^v þ4max 1r l r m ,O i A P9f x 00i ðl Þg 9in terms of (23),and bias v i of allcells in set P is equal to ^vþ.Step pute x 00i ðl Þ(O i A Q ,1r l r m )in set Q .Choose^vÀo Àmax 1r l r m ,O i A Q9f x 00i ðl Þg 9in terms of (24),and bias v i of all cells in set Q is equal to ^vÀ.Q.Han et al./Neurocomputing 97(2012)192–200195Step 11.Synthesize the CNN with the connection weight matrices A ,C ,D and bias vector V .5.Numerical examplesIn this section,we will give some numerical simulations to verify the theoretical results in this paper.Consider the same example introduced in [16].The inputs and the output patterns of a CNN are represented by two pairs of (5Â5)pixel images showed in Figs.1and 2(black pixel ¼1,white pixel ¼À1),where the inputs of the CNN compose the word ‘‘MO’’in Fig.2(a),and the patterns to be memorized to constitute the word ‘‘LS’’in Fig.2(b).We design all parameters of a CNN to realize associative memories for patterns in Fig.2by use of design procedure (i)in Section 4.Step 1.In terms of Fig.2,we get memory patterns G ¼ððÀ1,1,À1,...,À1ÞT ,ðÀ1,1,1,...,À1ÞT Þand inputs matrix U ¼ðð1,À1,À1,...,1ÞT ,ðÀ1,1,1,...,1ÞT Þ.Step 2.From memory patterns,all cells of the CNN can be divided into three sets,P ¼f O 2,O 7,O 12,O 22,O 23,O 24g ,Q ¼f O 1,O 5,O 6,O 8,O 9,O 10,O 11,O 15,O 16,O 18,O 20,O 21,O 25g and R ¼f O 3,O 4,O 13,O 14,O 17,O 19g .Step 3.Choose biases of all cells in set R are equal to zero,namely,v 3¼v 4¼v 13¼v 14¼v 17¼v 19¼0.Step 4.Let c ij ¼1,1r i r N ,1r j r M ,then we can obtain C ¼diag ð1,1,...,1Þn Ân .Step 5.Let L ¼diag ð4,4,...,4Þn Ân ,then we have L 0¼diag 4,4,...,4ðÞnm Ânm .Step 6.From Eq.(14),we get LA ¼ð0,0,0,0,2,0,0,0,0ÞT .Then,we have A ¼diag ð2Þn Ân .Step 7.From Eq.(18),we get LD ¼À1:5682,À2:9394,ðÀ0:6136,1:6515,0:6136,0:1212,0:3864,0:3258,0:25ÞT .Then,we can obtain D .Step 8.Let ^vþ¼20in set P .Therefore,v 2¼v 7¼v 12¼v 22¼v 23¼v 24¼20.Step 9.Let ^vÀ¼À20in set Q .Therefore,v 1¼v 5¼v 6¼v 8¼v 9¼v 10¼v 11¼v 15¼v 16¼v 18¼v 20¼v 21¼v 25¼À20.Step 10.Synthesize the CNN with A ,C ,D and V .Note that the a ii 4c ij in the example,however,in previouspaper [15],Pn j ¼19a ij þij 9o c ij must be satisfied.Therefore,in the paper,though the initial states of a CNN have to return to zero,wereduce many limitations for the relationship among cloning templates.From the above ten steps,we can get a CNN which can realize associative memories from ‘‘MO’’to ‘‘LS’’.In Fig.3,when the inputs of the CNN are ‘‘O’’,we can get time response curves of all cells in the CNN in Fig.3.In Fig.3,we find that states of all cells will be stable after a time.When states of all cells are stable,the value of equilibrium point isx n ¼À22:5671,20:7718,4:1968,3:9240,À19:7945,ðÀ20:2415,18:5446,À26:3927,À29:7259,À27:6048,À20:2415,19:7718,4:7119,3:2877,À24:4686,À20:2415,À3:7271,À18:1355,4:7119,À23:6959,À20:1657,20:3248,26:3700,28:2790,À21:1051ÞT :Therefore,we know that all outputs of the CNN corresponding to the equilibrium point aref ðx n Þ¼À1,1,1,1,À1,À1,1,À1,À1,À1,À1,1,1,1,À1,À1,ðÀ1,À1,1,À1,À1,1,1,1,À1ÞT ,where the outputs of the CNN are same with ‘‘S’’.When the inputs of the CNN are ‘‘M’’,we can get time response curves of all cells in the CNN in Fig.4.Then the value of equilibrium point isx n ¼À20:9309,23:3777,À4:0755,À3:9543,À22:3247,ðÀ23:5141,26:4382,À16:0296,À18:7113,À21:1657,Fig.1.Flow chart about how to get parameters of a CNN by use of Theorem 1and Corollary3.Fig.2.(a)Inputs of a CNN and (b)outputs of the CNN or memory patterns.Q.Han et al./Neurocomputing 97(2012)192–200196。
The Auslander-Bridger formula and the Gorenstein property for coherent rings
1. Introduction In addressing a problem posed by Glaz ([10], [11]), Hamilton and the second author give a definition of Cohen-Macaulay for commutative rings which agrees with the usual notion for Noetherian rings with the property that every coherent regular ring is CohenMacaulay [12]. (A quasi-local ring is defined to be regular if every finitely generated ideal has finite projective dimension.) A natural question is whether there is a reasonable concept of Gorenstein for commutative rings such that every coherent regular ring is Gorenstein and every coherent Gorenstein ring is Cohen-Macaulay. In this paper, we develop such a theory of coherent Gorenstein rings which mirrors much of the theory in the Noetherian case. Central to this development is the the concept of Gorenstein dimension (G-dimension, for short), first introduced in the context of finitely generated modules over Noetherian rings by Auslander and Bridger [1]. In particular, we prove the following generalization of the Auslander-Bridger formula for coherent rings using a notion of depth for arbitrary quasilocal rings developed by Barger [2], Hochster [13], and Northcott [1AS MARLEY
A Bell Theorem Without Inequalities for Two Particles, Using Efficient Detectors
2 reality, it is a very problematical state, since two independently produced particles that have never met are suddenly thrown into an entanglement, even though they have no shared history. We will show that the situation is strange enough so that one cannot reproduce the quantum perfect correlations of the entanglement-swapped state with a classical, deterministic theory. While the original experiment6 detected the cross-entangled pair of particles in a singlet state, in principle we can detect any of the four Bell states and
[JMAA384(2)(2011)683-689]Sums of weighted composition operators on H(infinity)
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Sums of weighted composition operators on H∞
Kei Ji Izuchi a,1, Shûichi Ohno b,∗,2
a Department of Mathematics, Niigata University, Niigata 950-2181, Japan b Nippon Institute of Technology, Miyashiro, Minami-Saitama 345-8501, Japan
Tight Bounds on the Capacity of Binary Input random CDMA Systems
a rXiv:083.1454v1[cs.IT]1Mar28Tight Bounds on the Capacity of Binary Input random CDMA Systems Satish Babu Korada and Nicolas Macris School of Information and Communication Sciences Ecole Polytechnique F´e d´e rale de Lausanne LTHC-IC-Station 14,CH-1015Lausanne Switzerland March 10,2008Abstract We consider multiple access communication on a binary input additive white Gaussian noise channel using randomly spread code division.For a general class of symmetric distributions for spreading coefficients,in the limit of a large number of users,we prove an upper bound on the capacity,which matches a formula that Tanaka obtained by using the replica method.We also show concentration of various relevant quantities including mutual information,capacity and free energy.The mathe-matical methods are quite general and allow us to discuss extensions to other multiuser scenarios.1Introduction Code Division Multiple Access (CDMA)has been a successful scheme for reliable communication between multiple users and a common receiver.The scheme consists of K users modulating their information sequence by a signature sequence,also known as spreading sequence,of length N and transmitting.The number N is sometimes referred to as the spreading gain or the number of chips per sequence.The receiver obtains the sum of all transmitted signals and the noise which is assumed to be white and Gaussian (AWGN).The achievable rate region (for real valued inputs)with power constraints and optimal decoding has been given in [1].There it is shown that the achievable rates depend only on the correlation matrix of the spreading coefficients.It is well known that these detectors have exponential (in K )complexity.Therefore,it is important to analyze the performance under sub-optimal but low-complexity detectors like the linear detectors.For a good overview of these detectors we refer to [2].In [3],the authorsconsidered random spreading (spreading sequences are chosen randomly)and analyzed the spectral efficiency,defined as the bits per chip that can be reliably transmitted,for these detectors.In the large-system limit (K →∞,N →∞,KOur main contributions in this paper are twofold.First we prove that Tanaka’s formula is an upper bound to the capacity for all values of the parameters and second we prove various useful concentration theorems in the large-system limit.1.1Statistical Mechanics ApproachThere is a natural connection between various communication systems and statistical mechanics of random spin systems,stemming from the fact that often in both systems there is a large number of degrees of freedom(bits or spins),interacting locally,in a random environment.So far,there have been applications of two important but somewhat complementary approaches of statistical mechanics of random systems.Thefirst one is the very important but mathematically uncontrolled replica method.The merit of this approach is to obtain conjectural but rather explicit formulas for quantities of interest such as, free energy,conditional entropy or error probability.In some cases the naturalfixed point structure embodied in the meanfield formulas allows to guess good iterative algorithms.This program has been carried out for linear error correcting codes,source coding,multiuser settings like broadcast channel(see for example[11],[12],[13])and the case of interest here[7]:randomly spread CDMA with binary inputs.The second type of approach aims at a rigorous understanding of the replica formulas and has its origins in methods stemming from mathematical physics(see[14,15],[9]).For systems whose underlying degrees of freedom have Gaussian distribution(Gaussian input symbols or Gaussian spins in continuous spin systems)random matrix methods can successfully be employed.However when the degrees of freedom are binary(binary information symbols or Ising spins)these seem to fail,but the recently developed interpolation method[14],[15]has had some success1.The basic idea of the interpolation method is to study a measure which interpolates between the posterior measure of the ideal decoder and a meanfield measure.The later can be guessed from the replica formulas and from this perspective the replica method is a valuable tool.So far this program has been developed only for linear error correcting codes on sparse graphs and binary input symmetric channels[16],[17].In this paper we develop the interpolation method for the random CDMA system with binary inputs (in the large-system limit).The situation is qualitatively different than the ones mentioned above in that the“underlying graph”is complete.Superficially one might think that it is similar to the Sherrington-Kirkpatrick model which was thefirst one treated by the interpolation method.However as we will see the analysis of the randomly spread CDMA system is substantially different due to the structure of the interaction between degrees of freedom.1.2Communication SetupWe consider a scenario where K users send binary information symbols xk =(s1k,...,s Nk)t where the components are independently identically distributed.For each timedivision(or chip)interval i=1,...,N the received signal y√=(n1,...,n N)t are independent identically distributed Gaussian variables N(0,1)so that the noise power isσ2.The variance of s ik is set to1and the scaling factor1/√1Let us point out that,as will be shown later in this paper,the interpolation method can also serve as an alternative to random matrix theory for Gaussian inputs.In particular,our favorite Gaussian and binary cases are included in this class,and also any compactly supported distribution.An inspection of our proofs suggests that the results could be extended to a larger class satisfying:Assumption B.The distribution p(s ik)is symmetric withfinite second and fourth moments. However to keep the proofs as simple as possible only one of the theorems is proven with such generality.In the sequel we use the notations s for the N×K matrix(s ik),S for the corresponding random matrix,and X for the input and output random vectors.Our main interest is in proving a“tight”upper bound onC K=1E S[I(X)](1)in the large-system limit K→+∞with K(x;Yand thus so is its average.Moreover the later is invariant under the transformations p X(ǫ1x1,ǫ2x2,...,ǫK x K)whereǫi=±bining these two facts we deduce that the maximum in(1)is attained for the convex combination1(ǫ1x1,...,ǫK x K)=1KmaxQ Kk=1p k(x k)I(X)(2)where the maximum is over p i(x)=p iδ(x−1)+(1−p i)δ(x+1)and p i∈[0,1],i=1,...,k.In the large-system limit we are able to prove a concentration theorem for the mutual information I(X) which implies that if(p1,...,p K)belongs to afinite discrete set D with cardinality increasing at most polynomially in K,then(2)concentrates on1;Y2as long as1;YThen,by the analysis in[18],formula(1)gives the capacity.If users do not cooperate p X| Y|s[H(X)]is the average over Y|y(xZ(y2σ2 y2s x,s)= x(x2σ2 y2s xis carried out with the distribution induced by the channel transition probabilityp(y0p X(x2σ2y2s x(√(√,s)(5)where in the sum x.In view of this it is not surprising that the free energyf(yKln Z(yK I(X)=−1|s[f(y2β−min p X,S[f(yis attained for p X)=12(1+m)−1λz+λ))(10)withλ=12 2πdz,has to be maximized over a parameter2m.It iseasy3to see that the maximizer must satisfy thefixed point conditionm= Dz tanh(√2this parameter can be interpreted as the expected value of the MMSE estimate for the information bits 3using integration by parts formula for Gaussian random variablesThe formal calculations involved in the replica method make clear that the formula(9)should not depend on the distribution of the spreading sequence(see[7]).In the present problem one expects a priori that replica symmetry is not broken because of a gauge symmetry induced by channel symmetry.For this reason Tanaka’s formula is conjectured to be exact. Our upper bound(Theorem6)on the capacity precisely coincides with the above formulas and strongly supports this conjecture.Recent work announced by Montanari and Tse[10]also provides strong support to the conjecture at least in a regime ofβwithout phase transitions(more precisely,forβ≤βs(σ)whereβs(σ)is the maximal value ofβsuch that the solution of(12)remains unique).The authorsfirst solve the case of sparse signature sequence(using the area theorem and the data processing inequality)in the limit K→∞.Then the dense signature sequence(which is of interest here)is recovered by exchanging the K→∞and sparse→dense limits.1.4Gaussian inputsIn the case of continuous inputs x k∈R,in formulas(4),(5) x.The capacity is maximized by a Gaussian prior,p X)=e−||x22log(1+σ−2−12βlog(1+σ−2β−18βσ−2(14)whereQ(x,z)= z)2+1− z)2+1 2On the other hand Tanaka applied the formal replica method to this case and found(9)withc RS(m)=12βlogλσ2−λ1+λ(16)Solving(16)we obtain m=σ2N fixed(Theorems1,3in section2.1).As we will see the mathematical underpinning of this is the concentration of a more fundamental object,namely,the“free energy”of the associated spin system(Theorem2).Infact this turns out to be important in the proof of the bound on capacity.When the spreading coefficients are Gaussian the main tool used is a powerful theorem[9]of the concentration of Lipschitz functions of many independent Gaussian variables,and this leads to subexponential concentration bounds.For more general spreading coefficient distributions such tools do not suffice and we have to combine them with martingale arguments which lead to weaker algebraic bounds.Since the concentration proofs are mainly technical they are presented in appendices B,C.Sections3and4form the core of the paper.They detail the proof of the main Theorem6announced in section2.4,namely the tight upper bound on capacity.We use ideas from the interpolation method combined with a non-trivial concentration theorem for the empirical average of soft bit estimates.Section5shows that the average capacity is independent of the spreading sequence distribution at least for the case where it is symmetric and decays fast enough(Theorem4in section2.2).This enables us to restrict ourselves to the case of Gaussian spreading sequences which is more amenable to analysis. The existence of the limit K→∞for the capacity is shown in section6.Section7discusses various extensions of this work.We sketch the treatment for unequal powers for each user as well as colored noise.As alluded to before the bound on capacity for the case of Gaussian inputs can also be obtained by the present method and we give some indications to this effect.The appendices contain the proofs of various technical calculations.Preliminary versions of the results obtained in this paper have been summarized in references[20]and[21].2Main Results2.1ConcentrationIn the case of a Gaussian input signal,the concentration can be deduced from general theorems on the concentration of the spectral density for random matrices,but this approach breaks down for binary inputs.Here we prove,Theorem1(concentration of capacity,Gaussian spreading sequence,binary inputs).Assume the distribution p(s ik)are standard Gaussians.Givenǫ>0,there exists an integer K1=O(|lnǫ|) independent of p X;Y;Yσ4(64β+32+σ2)−1.16The mathematical underpinning of this result is in fact a more general concentration result for the free energy(6),that will be of some use latter on.Theorem2(concentration of free energy,Gaussian spreading sequence,binary inputs.). Assume the distribution p(s ik)are standard Gaussians.Givenǫ>0,there exists an integer K2= O(|lnǫ|)independent of p X√,s)−E Y,s)]|≥ǫ]≤3e−α2ǫ2σ4β3β+σ)−2.32We prove these theorems thanks to powerful probabilistic tools developed by Ledoux and Talagrand for Lipschitz functions of many Gaussian random variables.These tools are briefly reviewed in Appendix B for the convenience of the reader and the proofs of the theorems are presented in Appendix C. Unfortunately the same tools do not apply directly to the case of other spreading sequences.However in this case the following weaker result can at least be obtained.Theorem3(concentration,general spreading sequence).Assume the spreading sequence satisfies assumption B.There exists an integer K1independent of p X;Y;YKǫ2P[f(y,S[f(yKǫ2for some constantα>0and independent of K.To prove such estimates it is enough(by Chebycheff)to control second moments.For the mutual information we simply have to adapt martingale arguments of Pastur,Scherbina and Tirrozzi,[22,23] whereas the case of free energy is more complicated because of the additional Gaussian noisefluctuations. We deal with these by combining martingale arguments and Lipschitz function techniques.The concentration of capacity,namelyP[|maxp X ;Y E S[I(X)]|≥ǫK]≤α)P[maxp X ;Y;YKǫ2(18)To see this it suffices to note that for two positive functions f and g we have|max f−max g|≤max|f−g|. But unfortunately it is not clear how to extend our proofs to obtain(18).However as announced in the introduction we can deduce(18)from our theorems,by using the union bound,as long as the maximum is carried out over afinite set(sufficiently small with respect to K)of distributions.We wish to argue here that Theorem2suggests a method for proving the concentration of the bit error rate(BER)for uncoded communication1KKk=1x0,kˆx k)(19)where the MAP bit estimate for uncoded communication is defined through the marginal of(3),namelyˆx k=argmax xk ={±1}p(x k|yx k p(x,s)(a soft bit estimate or“magnetization”)can be obtained from the free energy by addingfirst an in-finitesimal perturbation(“small external magneticfield”)to the exponent in(3),namely h K k=1x0k x k, and then differentiating the perturbed free energy4,1dh 1,s)However one really needs to relate sign x k to the derivative of the free energy and this does not appear to be obvious.One way out is to introduce product measures of n copies(also called“real replicas”)of the posterior measurep(x,s)p(x,s)...p(x,s)and then relateKk=1(x0k x k )n=K k=1 x0k x1k...x0k x n k nto a suitable derivative of the replicated free energy.Then from the set of all moments one can in principle reconstruct sign x k .Thus one could try to deduce the concentration of the BER from the one for the free energy.However the completion of this program requires a uniform,with respect the system size,control of the derivative of the free energy precisely at h=0,which at the moment is still lacking5.2.2Independence with respect to the distribution of the spreading sequence The replica method leads to the same Tanaka formula for general class of symmetric distributions p(s ik)=p(−s ik).We are able to prove this:in particular binary and Gaussian spreading sequences lead to the same capacity.Theorem4.Consider CDMA with binary inputs and assume A for the spreading sequence.Let C g be the capacity for Gaussian spreading sequences(symmetric i.i.d with unit variance).Thenlim K→+∞(C K−C g)=0This theorem turns out to be very useful in order to obtain the bound on capacity because it allows us to make use of convenient integration by parts identities that have no clear counterpart in the non-Gaussian case.The proof of the theorem is given in section5.2.3Existence of the limit K→+∞The interpolation method can be used to show the existence of the limit K→+∞for C K.Theorem5.Consider CDMA with binary inputs and assume A for the spreading sequences with uniform input distribution.ThenlimK→∞C K exists(20)The proof of this theorem is given in section6for Gaussian spreading sequences.The general case then follows because of Theorem4.2.4Tight upper bound on the capacityThe main result of this paper is that Tanaka’s formula(10)is an upper bound to the capacity for all values ofβ.Theorem6.Consider CDMA with binary inputs and assume A for the spreading sequence.We havelim K→∞C K≤minm∈[0,1]c RS(m)(21)where c RS(m)is given by(10).If we combine this result with an inequality in Montanari and Tse[10],and exchanging as they do the limits of K→+∞and sparse→dense,one can deduce that the equality holds for some regime of noise smaller than a critical value.This value corresponds to the threshold for belief propagation decoding.Note that this equality is valid even ifβis such that there is a phase transition(thefixed point equation(12)has many solutions),whereas in[10]the equality holds for values ofβfor which the phase transition does not occur.Since the proof is rather complicated wefind it useful to give the main ideas in an informal way.The integral term in(10)suggests that we can replace the original system with a simpler system where the user bits are sent through K independent Gaussian channels given by˜y k=x k+1λw k(22)where w k∼N(0,1)andλis an effective SNR.Of course this argument is a bit naive because this effective system does not account for the extra terms in(10),but it has the merit of identifying the correct interpolation.We introduce an interpolating parameter t∈[0,1]such that the independent Gaussian channels correspond to t=0and the original CDMA system corresponds to t=1(see Figure2.4)It is convenient to denote the SNR of the original Gaussian channel as B(that is B=σ−2).Then(11)becomesλ=Bsx t ))yλ(t ))N (0,λ(t ))˜y ˜y ˜y Figure 1:The information bits x k are transmitted through the normal CDMA channel with variance 1λ(t )We introduce two interpolating SNR functions λ(t )and B (t )such thatλ(0)=λ,B (0)=0and λ(1)=0,B (1)=B (23)andB (t )1+βB (1−m )(24)The meaning of (24)is the following.In the interpolating t -system the effective SNR seen by each user has an effective t -CDMA part and an independent channel part λ(t )chosen such that the total SNR is fixed to the effective SNR of the CDMA system.There is a whole class of interpolating functions satisfying the above conditions but it turns out that we do not need to specify them more precisely except for the fact that B (t )is increasing,λ(t )is decreasing and with continuous first derivatives.Subsequent calculations are independent of the particular choices of functions.The parameter m is to be considered as fixed to any arbitrary value in [0,1].All the subsequent calculations are independent of its value,which is to be optimized to tighten the final bound.We now have two sets of channel outputs y(from the independent channels with noise variance λ(t )−1)and the interpolating communication system has a posterior distributionp t (x ,˜y 2K Z (y ,s )exp −B (t )−N −1 2−λ(t )−x (X2K .By analyzing the mutual information E S [I t (X ,˜Y ;Y ;˜Y,˜Y ,˜y 2K x ( 2πλ(t )−1)K e −B (t )−N −10 2−λ(t )−xIn order to carry out this program successfully it turns out that we need a concentration result on empirical average of the“magnetization”,m1=1,˜y0is transmitted.The distribution of the received vectors with this assumption isp t(y|s)=12πB(t)−1)N( 2 y2s x2 ˜y0 2(27)For technical reasons that will become clear only in the next section we consider a slightly more general interpolation system where the perturbation termh u(x uKk=1h k x k+u K k=1x0k x k−√=B(t)−1/2n0and˜y+x|n,h Zt,uexp −1+N−12s(x) 2(29)−1+λ(t)10−x)with the obvious normalization factor Z t,u.We define a free energyf t,u(n,hKln Z t,u(30) For t=1we recover the original free energy,E[f(y2+limu→0E[f1,u(n,hwhile for t=0the statistical sums decouple and we have the explicit result61,w,s)]=−1λz+λ))(31)where E denotes the appropriate collective expectation over random objects.In view of formula(7)in order to obtain the average capacity it is sufficient to computelimK→+∞limu→0E[f1,u(n,h2(32)There is no loss in generality in settingx0k=1(33) for the input symbols.From now on in sections3,4,and6we stick to(33).We also use the shorthand notationsz k=x0k−x k=1−x k,f t,u(n,h)|≤2√u E[|h k|]+u(34) therefore we can permute the two limits in(32)and computelimu→0limK→+∞E[f1,u]+1dtE[f t,u](35) Our task is now reduced to estimatinglimu→0limK→+∞1dtdKKk=1x k(36) A closely related quantity is the“overlap parameter”q12=1(1)|n,h(2)|n,h,w,s).u),uniformly in KLemma1.The distributions of m1and q12defined asP m1(x)=E δ(x−m1) t,u,P q12(x)=E δ(x−q12) t,uare equal,namelyP m1(x)=P q12(x)In particular the following identity holdsE[ m1 t,u]=E[ q12 t,u](38) Such identities are known as Nishimori identities in the statistical physics literature and are a consequence of a gauge symmetry satisfied by the measure E − t,u.They have also been used in the context of communications(see[11],[16]).For completeness a sketch of the proof is given in Appendix F.The next two identities also follow from similar considerations.Lemma2.LetZ+Ns z(α),α=1,2corresponding to z(α)k =1−x(α)k.We have then12 t,u]=1(39) andE[ (n(2))(z(2)) t,u]= k E[ (n)z k t,u](40)3.3Concentration of MagnetizationA crucial feature of the calculation in the next paragraph is that m1(and q12)concentrate,namely Theorem7.Fix anyǫ>0.For Lebesgue almost every u>ǫ,lim N→∞ 1dt E |m1−E m1 t,u| t=0The proof of this theorem,which is the point where the careful tuning of the perturbation is needed, has an interest of its own and is presented section4.Similar statements in the spin glass literature have been obtained by Talagrand[9].The usual signature of replica symmetry breaking is the absence of concentration for the overlap parameter q12.This theorem combined with the Nishimori identity “explains”why the replica symmetry is not broken.We will also need the following corollaryCorollary1.The following holds1·s z N3/2E n t,u(1−E m1 t,u)+o N(1)with lim N→+∞o N(1)=0for almost every u>0.Proof.By the Cauchy-Schwartz inequality1·s z N3/2(E (n)2 t,u)1/2×(E (E m1 t,u−m1)2 t,u)1/2Because of the concentration of the magnetization m1(theorem7)it suffices to prove thatE N−33.4Computation ofddtE [f t,u ]=T 1+T 2(42)whereT 1=−λ′(t )λ(t )KE wt,u −λ′(t )·zK√2·s z2+)·z 2(1)·(wλ(t )z2KE zt,u=−λ′(t )2E 1−m 1 t,uTo obtain the second equality we remark that the w2(1+β(1−m )B (t ))2E 1−m 1 t,u(45)3.4.2Transforming T 2The term T 2can be rewritten asT 2=−B ′(t ) 2t,u +B ′(t )2+B ′(t )B (t )K√·s z2NE nt,u (46)Now we use integration by parts with respect to s ik ,T 2=−B ′(t )·Z·z2KNE (n(2))(z(2)) t,uand the Nishimori identity (40)T 2=−B ′(t )·Z212 z k t,u ]−B ′(t ) 2KN 3/2kE (n)(1−x k ) t,uSince12=12βE 1−m 1 t,u +o N (1)−βB ′(t ) 2KN 1/2E (n )(1−m 1) t,uApplying Corollary 1to the last expression for T 2together with (46)we obtain a closed affine equation for the later,whose solution isT 2=−B ′(t )E 1−m 1 t2βln(1+βB (1−m ))from (35)and use the integral representation12β1dtβB ′(t )(1−m )2βln(1+βB (1−m ))+1dtd2(1+βB (t )(1−m ))If one uses (42)and expressions (45),(47)some remarkable algebra occurs in the last integral.The integrand becomesR (t )+B ′(t )(1−m )2(1+βB (t )(1−m ))2(1+βB (t )E 1−m 1 t,u )So the integral has a positive contribution 10dtR (t )≥0plus a computable contribution equal to B (1−m )2(1−m ).Finally thanks to (31)we find1λz +λ))−12βln(1+βB (1−m ))−λu )(48)where for a.e u >ǫ,lim N →∞o N (1)=0.We take first the limit N →∞,then u →ǫ(along some appropriate sequence)and then ǫ→0to obtain a formula for the free energy where the only non-explicit contribution is 10dtR (t ).Since this is positive for all m ,we obtain a lower bound on the free energy which is equivalent to the announced upper bound on the capacity.4Concentration of MagnetizationThe goal of this section is to prove Theorem 7.The proof is organized in a succession of lemmas.By the same methods used for Theorem 2we can proveLemma 3.There exists a strictly positive constant α(which remains positive for all t and u )such thatP [|f t,u −E [f t,u ]|≥ǫ]=O (e−αǫ2√Lemma 4.When considered as a function of u ,f t,u is convex in u .Proof.We simply evaluate the second derivative and show it is positive.d f t,u) t,u −1uk|h k |where we have definedL (xK1ukh k x k +1du 2=14u 3/2kh k x kt,u+1)2 t,u − L (x)turns out to be very useful and satisfies two concentration properties.Lemma 5.For any a >ǫ>0fixed,a ǫdu EL (x) t,u t,u=O1KProof.From equation (49),wehavea ǫdu EL (x) t,u2 t,u≤aǫdu1du 2E [f t,u ]≤1duE [f t,a ]−dKIn the very last equality we use that the first derivative of E [f t,u ]is bounded for u ≥ǫ.Using Cauchy-Schwartz inequality for E − t,u we obtain the lemma.Lemma 6.For any a >ǫ>0fixed,a ǫdu EL (x) t,u=O116Proof.From convexity of f t,u with respect to u (lemma 4)we have for any δ>0,dduE [f t,u ]≤f t,u +δ−f t,udu E [f t,u ]≤f t,u +δ−E [f t,u +δ]δ+dduE [f t,u ]A similar lower bound holds with δreplaced by −δ.Now from Lemma 3we know that the first twoterms are O (K1KKk =1|h k |are O (1K )we get EL (x) t,u ≤1√δO14+dduE [f t,u ]We will choose δ=18.Note that we cannot assume that the difference of the two derivatives is smallbecause the first derivative of the free energy is not uniformly continuous in K (as K →∞it may develop jumps at the phase transition points).The freeenergy itself is uniformly continuous.For this reason if we integrate with respect to u,using (34)we geta ǫdu EL (x) t,u ≤O 116Using the two last lemmas we can prove Theorem 7.Proof of Theorem 7:Combining the concentration lemmas we geta ǫdu E |L (x) t,u | t,u ≤O 116For any function g (x)|≤1,we haveaǫdu |E L (x) t,u −E L (x) t,u | t,u ≤aǫdu E |L (x) t,u | t,uMore generally the same thing holds if one takes a function depending on many replicas such asg (x(2))=q ing integration by parts formula with respect to h k ,E L (x2K √2E (1+q 12)q 12 t,u −12E (1+q 12)q 12 t,u =1) t,u E q 12 t,u =12(E m 1 t +(E m 1 t )2)(51)From equations (50)and (51),we geta ǫdu |E m 21 t,u −(E m 1 t,u )2|≤O116Now integrating with respect to t and exchanging the integrals (by Fubini’s theorem),we geta ǫdu 10dt |E m 21 t,u −(E m 1 t,u )2|≤O116The limit of the left hand side as K →∞therefore vanishes.By Lebesgue’s theorem this limit can beexchanged with the u integral and we get the desired result.(Note that one can further exchange the limit with the t -integral and obtain that the fluctuations of m 1vanish for almost every (t,u )).5Proof of independence from spreading sequence distribution:Theorem 4We consider a communication system with spreading values r ik generated from a symmetric distribution with unit variance and satisfying assumption A.We compare the capacity of this system to the Gaussian N (0,1)case whose spreading sequence values are denoted by s ik .The comparison is done through an interpolating system with respect to the two spreading sequencesv ik (t )=√1−ts ik ,0≤t ≤1。
INSEARCHOFEXCELLENCE
IN SEARCH OF EXCELLENCEExcellence is a journey and not a destination. In science itimplies perpetual efforts to advance the frontiers of knowledge.This often leads to progressively increasing specialization andemergence of newer disciplines. A brief summary of salientcontributions of Indian scientists in various disciplines isintroduced in this section.92P U R S U I T A N D P R O M O T I O N O F S C I E N C EThe modern period of mathematics research in India started with Srinivasa Ramanujan whose work on analytic number theory and modular forms ishighly relevant even today. In the pre-Independence period mathematicians like S.S. Pillai,Vaidyanathaswamy, Ananda Rau and others contributed a lot.Particular mention should be made of universities in Allahabad, Varanasi, Kolkata,Chennai and Waltair and later at Chandigarh,Hyderabad, Bangalore and Delhi (JNU). The Department of Atomic Energy came in a big way to boost mathematical research by starting and nurturing the Tata Institute of Fundamental Research (TIFR), which, under the leadership of Chandrasekharan, blossomed into a great school of learning of international standard. The Indian Statistical Institute, started by P.C. Mahalanobis,made its mark in an international scene and continues to flourish. Applied mathematics community owes a great deal to the services of three giants Ñ N.R. Sen, B.R. Seth and P .L. Khastgir. Some of the areas in which significant contributions have been made are briefly described here.A LGEBRAOne might say that the work on modern algebra in India started with the beautiful piece of work in 1958 on the proof of SerreÕs conjecture for n =2. A particular case of the conjecture is to imply that a unimodular vector with polynomial entries in n vari-ables can be completed to a matrix of determinantone. Another important school from India was start-ed in Panjab University whose work centres around Zassanhaus conjecture on groupings.A LGEBRAIC G EOMETRYThe study of algebraic geometry began with a seminal paper in 1964 on vector bundles. With further study on vector bundles that led to the mod-uli of parabolic bundles, principle bundles, algebraic differential equations (and more recently the rela-tionship with string theory and physics), TIFR has become a leading school in algebraic geometry. Of the later generation, two pieces of work need special mention: the work on characterization of affine plane purely topologically as a smooth affine surface, sim-ply connected at infinity and the work on Kodaira vanishing. There is also some work giving purely algebraic geometry description of the topologically invariants of algebraic varieties. In particular this can be used to study the Galois Module Structure of these invariants.L IE T HEORYThe inspiration of a work in Lie theory in India came from the monumental work on infinite dimensional representation theory by Harish Chandra, who has, in some sense, brought the sub-ject from the periphery of mathematics to centre stage. In India, the initial study was on the discrete subgroups of Lie groups from number theoretic angle. The subject received an impetus after an inter-national conference in 1960 in TIFR, where the lead-ing lights on the subject, including A. Selberg partic-M ATHEMATICAL S CIENCESC H A P T E R V I Iipated. Then work on rigidity questions was initiat-ed. The question is whether the lattices in arithmetic groups can have interesting deformations except for the well-known classical cases. Many important cases in this question were settled.D IFFERENTIALE QUATIONA fter the study of L-functions were found to beuseful in number theory and arithmetic geome-try, it became natural to study the L-functions arising out of the eigenvalues of discrete spectrum of the dif-ferential equations. MinakshisundaramÕs result on the corresponding result for the differential equation leading to the Epstein Zeta function and his paper with A. Pleijel on the same for the connected com-pact Riemanian manifold are works of great impor-tance. The idea of the paper (namely using the heat equation) lead to further improvement in the hands of Patodi. The results on regularity of weak solution is an important piece of work. In the later 1970s a good school on non-linear partial differential equa-tions that was set up as a joint venture between TIFR and IISc, has come up very well and an impressive lists of results to its credit.For differential equations in applied mathematics, the result of P.L. Bhatnagar, BGK model (by Bhatnagar, Gross, Krook) in collision process in gas and an explanation of Ramdas Paradox (that the temperature minimum happens about 30 cm above the surface) will stand out as good mathematical models. Further significant contributions have been made to the area of group theoretic methods for the exact solutions of non-liner partial differential equations of physical and engineering systems.E RGODIC T HEORYE arliest important contribution to the Ergodic the-ory in India came from the Indian Statistical Institute. Around 1970, there was work on spectra of unitary operators associated to non-singular trans-formation of flows and their twisted version, involv-ing a cocycle.Two results in the subjects from 1980s and 1990s are quoted. If G is lattice in SL(2,R) and {uÐt} a unipotent one parameter subgroup of G, then all non-periodic orbits of {uÐt} on GÐ1 are uniformly distributed. If Q is non-generate in definite quadratic form in n=variables, which is not a multiple of rational form, then the number of lattice points xÐwith a< ½Q(x)½< b, ½½x½½< r, is at least comparable to the volume of the corresponding region.N UMBER T HEORYT he tradition on number theory started with Ramanujan. His work on the cusp form for the full modular group was a breakthrough in the study of modular form. His conjectures on the coefficient of this cusp form (called RamanujanÕs tau function) and the connection of these conjectures with conjectures of A. Weil in algebraic geometry opened new research areas in mathematics. RamanujanÕs work (with Hardy) on an asymptotic formula for the parti-tion of n, led a new approach (in the hands of Hardy-Littlewood) to attack such problems called circle method. This idea was further refined and S.S. Pillai settled WaringÕs Conjecture for the 6th power by this method. Later the only remaining case namely 4th powers was settled in mid-1980s. After Independence, the major work in number theory was in analytic number theory, by the school in TIFR and in geometry of numbers by the school in Panjab University. The work on elliptic units and the con-struction of ray class fields over imaginary quadratic fields of elliptic units are some of the important achievements of Indian number theory school. Pioneering work in BakerÕs Theory of linear forms in logarithms and work on geometry of numbers and in particular the MinkowskiÕs theorem for n = 5 are worth mentioning.P ROBABILITY T HEORYS ome of the landmarks in research in probability theory at the Indian Statistical Institute are the following:93 P U R S U I T A N D P R O M O T I O N O F S C I E N C Eq A comprehensive study of the topology of weak convergence in the space of probability measures on topological spaces, particularly, metric spaces. This includes central limit theorems in locally compact abelian groups and Milhert spaces, arithmetic of probability distributions under convolution in topological groups, Levy-khichini representations for characteristic functions of probability distributions on group and vector spaces.q Characterization problems of mathematical statistics with emphasis on the derivation of probability laws under natural constraints on statistics evaluated from independent observations.q Development of quantum stochastic calculus based on a quantum version of ItoÕs formula for non-commutative stochastic processes in Fock spaces. This includes the study of quantum stochastic integrals and differential equations leading to the construction of operator Markov processes describing the evolution of irreversible quantum processes.q Martingale methods in the study of diffusion processes in infinite dimensional spaces.q Stochastic processes in financial mathematics.C OMBINATORICST hough the work in combinatorics had been ini-tiated in India purely through the efforts of R.C.Bose at the Indian Statistical Institute in late thirties, it reached its peak in late fifties at the University of North Carolina, USA, where he was joined by his former student S.S.Shrikhande. They provided the first counter-example to the celebrat-ed conjecture of Euler (1782) and jointly with Parker further improved it. The last result is regarded a classic.In the absence of these giants there was practically no research activity in this area in India. However, with the return of Shrikhande to India in 1960 activities in the area flourished and many notable results in the areas of embedding of residual designs in symmetric designs, A-design conjecture and t-designs and codes were reported.T HEORY OF R ELATIVITYI n a strict sense the subject falls well within the purview of physics but due to the overwhelming response by workers with strong foundation in applied mathematics the activity could blossom in some of the departments of mathematics of certain universities/institutes. Groups in BHU, Gujarat University, Ahmedabad, Calcutta University, and IIT, Kharagpur, have contributed generously to the area of exact solutions of Einstein equations of gen-eral relativity, unified field theory and others. However, one exact solution which has come to be known as Vaidya metric and seems to have wide application in high-energy astrophysics deserves a special mention.N UMERICAL A NALYSIST he work in this area commenced with an attempt to solve non-linear partial differential equations governing many a physical and engineering system with special reference to the study of Navier-Stabes equations and cross-viscous forces in non-Newtonian fluids. The work on N-S equation has turned out to be a basic paper in the sense that it reappeared in the volume, Selected Papers on Numerical Solution of Equations of Fluid Dynamics, Applied Mathematics, through the Physical Society of Japan. The work on non-Newtonian fluid has found a place in the most prestigious volume on Principles of Classical Mechanics & Field Theory by Truesdell and Toupin. The other works which deserve mention are the development of extremal point collocation method and stiffy stable method.A PPLIED M ATHEMATICST ill 1950, except for a group of research enthusi-asts working under the guidance of N.R.Sen at Calcutta University there was practically no output in applied mathematics. However, with directives from the centre to emphasize on research in basic94P U R S U I T A N D P R O M O T I O N O F S C I E N C Eand applied sciences and liberal central fundings through central and state sponsored laboratories, the activity did receive an impetus. The department of mathematics at IIT, Kharagpur, established at the very inception of the institute of national importance in 1951, under the dynamic leadership of B.R.Seth took the lead role in developing a group of excellence in certain areas of mathematical sciences. In fact, the research carried out there in various disciplines of applied mathematics such as elasticity-plasticity, non-linear mechanics, rheological fluid mechanics, hydroelasticity, thermoelasticity, numerical analysis, theory of relativity, cosmology, magneto hydrody-namics and high-temperature gasdynamics turned out to be a trend setting one for other IITs, RECs, other Technical Institutes and Universities that were in the formative stages. B.R. SethÕs own researches on the study of Saint-VenamtÕs problem and transi-tion theory to unify elastic-plastic behaviour of mate-rials earned him the prestigious EulerÕs bronze medal of the Soviet Academy of Sciences in 1957. The other areas in which applied mathematicians con-tributed generously are biomechanics, CFD, chaotic dynamics, theory of turbulence, bifurcation analysis, porous media, magnetics fluids and mathematicalphysiology.95 P U R S U I T A N D P R O M O T I O N O F S C I E N C E。
协作移动机器人-前因和方向外文文献翻译、中英文翻译、外文翻译
Cooperative Mobile Robotics: Antecedents and DirectionsY. UNY CAOComputer Science Department, University of California, Los Angeles, CA 90024-1596ALEX S. FUKUNAGAJet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109-8099ANDREW B. KAHNGComputer Science Department, University of California, Los Angeles, CA 90024-1596Editors: R.C. Arkin and G.A. BekeyAbstract. There has been increased research interest in systems composed of multiple autonomous mobile robots exhibiting cooperative behavior. Groups of mobile robots are constructed, with an aim to studying such issues as group architecture, resource conflict, origin of cooperation, learning, and geometric problems. As yet, few applications of cooperative robotics have been reported, and supporting theory is still in its formative stages. In this paper, we give a critical survey of existing works and discuss open problems in this field, emphasizing the various theoretical issues that arise in the study of cooperative robotics. We describe the intellectual heritages that have guided early research, as well as possible additions to the set of existing motivations.Keywords: cooperative robotics, swarm intelligence, distributed robotics, artificial intelligence, mobile robots, multiagent systems1. PreliminariesThere has been much recent activity toward achieving systems of multiple mobile robots engaged in collective behavior. Such systems are of interest for several reasons:•tasks may be inherently too complex (or im-possible) for a single robot to accomplish, or performance benefits can be gained from using multiple robots;•building and using several simple robots can be easier, cheaper, more flexible and more fault-tolerant than having a single powerful robot foreach separate task; and•the constructive, synthetic approach inherent in cooperative mobile robotics can possibly∗This is an expanded version of a paper which originally appeared in the proceedings of the 1995 IEEE/RSJ IROS conference. yield insights into fundamental problems in the social sciences (organization theory, economics, cognitive psychology), and life sciences (theoretical biology, animal ethology).The study of multiple-robot systems naturally extends research on single-robot systems, butis also a discipline unto itself: multiple-robot systems can accomplish tasks that no single robot can accomplish, since ultimately a single robot, no matter how capable, is spatially limited. Multiple-robot systems are also different from other distributed systems because of their implicit “real-world” environment, which is presumably more difficult to model and reason about than traditional components of distributed system environments (i.e., computers, databases, networks).The term collective behavior generically denotes any behavior of agents in a system having more than one agent. the subject of the present survey, is a subclass of collective behavior that is characterized by cooperation. Webster’s dictionary [118] defines “cooperate” as “to associate with anoth er or others for mutual, often economic, benefit”. Explicit definitions of cooperation in the robotics literature, while surprisingly sparse, include:1. “joint collaborative behavior that is directed toward some goal in which there is a common interest or reward” [22];2. “a form of interaction, usually based on communication” [108]; and3. “[joining] together for doing something that creates a progressive result such as increasing performance or saving time” [137].These definitions show the wide range of possible motivating perspectives. For example, definitions such as (1) typically lead to the study of task decomposition, task allocation, and other dis-tributed artificial intelligence (DAI) issues (e.g., learning, rationality). Definitions along the lines of (2) reflect a concern with requirements for information or other resources, and may be accompanied by studies of related issues such as correctness and fault-tolerance. Finally, definition (3) reflects a concern with quantified measures of cooperation, such as speedup in time to complete a task. Thus, in these definitions we see three fundamental seeds: the task, the mechanism of cooperation, and system performance.We define cooperative behavior as follows: Given some task specified by a designer, a multiple-robot system displays cooperative behavior if, due to some underlying mechanism (i.e., the “mechanism of cooperation”), there is an increase in the total utility of the system. Intuitively, cooperative behavior entails some type of performance gain over naive collective behavior. The mechanism of cooperation may lie in the imposition by the designer of a control or communication structure, in aspects of the task specification, in the interaction dynamics of agent behaviors, etc.In this paper, we survey the intellectual heritage and major research directions of the field of cooperative robotics. For this survey of cooperative robotics to remain tractable, we restrict our discussion to works involving mobile robots or simulations of mobile robots, where a mobile robot is taken to be an autonomous, physically independent, mobile robot. In particular, we concentrated on fundamental theoretical issues that impinge on cooperative robotics. Thus, the following related subjects were outside the scope of this work:•coordination of multiple manipulators, articulated arms, or multi-fingered hands, etc.•human-robot cooperative systems, and user-interface issues that arise with multiple-robot systems [184] [8] [124] [1].•the competitive subclass of coll ective behavior, which includes pursuit-evasion [139], [120] and one-on-one competitive games [12]. Note that a cooperative team strategy for, e.g., work on the robot soccer league recently started in Japan[87] would lie within our present scope.•emerging technologies such as nanotechnology [48] and Micro Electro-Mechanical Systems[117] that are likely to be very important to co-operative robotics are beyond the scope of this paper.Even with these restrictions, we find that over the past 8 years (1987-1995) alone, well over 200papers have been published in this field of cooperative (mobile) robotics, encompassing theories from such diverse disciplines as artificial intelligence, game theory/economics, theoretical biology, distributed computing/control, animal ethology and artificial life.We are aware of two previous works that have surveyed or taxonomized the literature. [13] is abroad, relatively succinct survey whose scope encompasses distributed autonomous robotic systems(i.e., not restricted to mobile robots). [50] focuses on several well-known “swarm” architectures (e.g., SWARM and Mataric’s Behavior-based architecture –see Section 2.1) and proposes a taxonomy to characterize these architectures. The scope and intent of our work differs significantly from these, in that (1) we extensively survey the field of co-operative mobile robotics, and (2) we provide a taxonomical organization of the literature based on problems and solutions that have arisen in the field (as opposed to a selected group of architectures). In addition, we survey much new material that has appeared since these earlier works were published.Towards a Picture of Cooperative RoboticsIn the mid-1940’s Grey Walter, along with Wiener and Shannon, studied turtle-like robots equipped wit h light and touch sensors; these simple robots exhibited “complex social behavior” in responding to each other’s movements [46]. Coordination and interactions of multiple intelligent agents have been actively studied in the field of distributed artificial intelligence (DAI) since the early 1970’s[28], but the DAI field concerned itself mainly with problems involving software agents. In the late 1980’s, the robotics research community be-came very active in cooperative robotics, beginning with projects such as CEBOT [59], SWARM[25], ACTRESS [16], GOFER [35], and the work at Brussels [151]. These early projects were done primarily in simulation, and, while the early work on CEBOT, ACTRESS and GOFER have all had physical implementations (with≤3 robots), in some sense these implementations were presented by way of proving the simulation results. Thus, several more recent works (cf. [91], [111], [131])are significant for establishing an emphasis on the actual physical implementation of cooperative robotic systems. Many of the recent cooperative robotic systems, in contrast to the earlier works, are based on a behavior-based approach (cf. [30]).Various perspectives on autonomy and on the connection between intelligence and environment are strongly associated with the behavior-based approach [31], but are not intrinsic to multiple-robot systems and thus lie beyond our present scope. Also note that a recent incarnation of CEBOT, which has been implemented on physical robots, is based on a behavior-based control architecture[34].The rapid progress of cooperative robotics since the late 1980’s has been an interplay of systems, theories and problems: to solve a given problem, systems are envisioned, simulated and built; theories of cooperation are brought from other fields; and new problems are identified (prompting further systems and theories). Since so much of this progress is recent, it is not easy to discern deep intellectual heritages from within the field. More apparent are the intellectualheritages from other fields, as well as the canonical task domains which have driven research. Three examples of the latter are:•Traffic Control. When multiple agents move within a common environment, they typically attempt to avoid collisions. Fundamentally, this may be viewed as a problem of resource conflict, which may be resolved by introducing, e.g., traffic rules, priorities, or communication architectures. From another perspective, path planning must be performed taking into con-sideration other robots and the global environment; this multiple-robot path planning is an intrinsically geometric problem in configuration space-time. Note that prioritization and communication protocols – as well as the internal modeling of other robots – all reflect possible variants of the group architecture of the robots. For example, traffic rules are commonly used to reduce planning cost for avoiding collision and deadlock in a real-world environment, such as a network of roads. (Interestingly, behavior-based approaches identify collision avoidance as one of the most basic behaviors [30], and achieving a collision-avoidance behavior is the natural solution to collision avoidance among multiple robots. However, in reported experiments that use the behavior-based approach, robots are never restricted to road networks.) •Box-Pushing/Cooperative Manipulation. Many works have addressed the box-pushing (or couch-pushing) problem, for widely varying reasons. The focus in [134] is on task allocation, fault-tolerance and (reinforcement) learning. By contrast, [45] studies two boxpushing protocols in terms of their intrinsic communication and hardware requirements, via the concept of information invariants. Cooperative manipulation of large objects is particularly interesting in that cooperation can be achieved without the robots even knowing of each others’ existence [147], [159]. Other works in the class of box-pushing/object manipulation include [175] [153] [82] [33] [91] [94] [92][114] [145] [72] [146].•Foraging. In foraging, a group of robots must pick up objects scattered in the environment; this is evocative of toxic waste cleanup, harvesting, search and rescue, etc. The foraging task is one of the canonical testbeds for cooperative robotics [32] [151] [10] [67] [102] [49] [108] [9][24]. The task is interesting because (1) it can be performed by each robot independently (i.e., the issue is whether multiple robots achieve a performance gain), and (2) as discussed in Section 3.2, the task is also interesting due to motivations related to the biological inspirations behind cooperative robot systems. There are some conceptual overlaps with the related task of materials handling in a manufacturing work-cell [47]. A wide variety of techniques have been applied, ranging from simple stigmergy (essentially random movements that result in the fortuitous collection of objects [24] to more complex algorithms in which robots form chains along which objects are passed to the goal [49].[24] defines stigmergy as “the production of a certain behaviour in agents as a consequence of the effects produced in the local environment by previous behaviour”. This is actually a form of “cooperation without communication”, which has been the stated object of several for-aging solutions since the corresponding formulations become nearly trivial if communication is used. On the other hand, that stigmergy may not satisfy our definition of cooperation given above, since there is no performance improvement over the “naive algorithm” –in this particular case, the proposed stigmergic algorithm is the naive algorithm. Again, group architecture and learning are major research themes in addressing this problem.Other interesting task domains that have received attention in the literature includemulti-robot security systems [53], landmine detection and clearance [54], robotic structural support systems (i.e., keeping structures stable in case of, say ,an earthquake) [107], map making [149], and assembly of objects using multiple robots [175].Organization of PaperWith respect to our above definition of cooperative behavior, we find that the great majority of the cooperative robotics literature centers on the mechanism of cooperation (i.e., few works study a task without also claiming some novel approach to achieving cooperation). Thus, our study has led to the synthesis of five “Research Axes” which we believe comprise the major themes of investigation to date into the underlying mechanism of cooperation.Section 2 of this paper describes these axes, which are: 2.1 Group Architecture, 2.2 Resource Conflict, 2.3 Origin of Cooperation, 2.4 Learning, and 2.5 Geometric Problems. In Section 3,we present more synthetic reviews of cooperative robotics: Section 3.1 discusses constraints arising from technological limitations; and Section 3.2discusses possible lacunae in existing work (e.g., formalisms for measuring performance of a cooperative robot system), then reviews three fields which we believe must strongly influence future work. We conclude in Section 4 with a list of key research challenges facing the field.2. Research AxesSeeking a mechanism of cooperation may be rephrased as the “cooperative behavior design problem”: Given a group of robots, an environment, and a task, how should cooperative behavior arise? In some sense, every work in cooperative robotics has addressed facets of this problem, and the major research axes of the field follow from elements of this problem. (Note that certain basic robot interactions are not task-performing interactions per se, but are rather basic primitives upon which task-performing interactions can be built, e.g., following ([39], [45] and many others) or flocking [140], [108]. It might be argued that these interactions entail “control and coordination” tasks rather than “cooperation” tasks, but o ur treatment does not make such a distinction).First, the realization of cooperative behavior must rely on some infrastructure, the group architecture. This encompasses such concepts as robot heterogeneity/homogeneity, the ability of a given robot to recognize and model other robots, and communication structure. Second, for multiple robots to inhabit a shared environment, manipulate objects in the environment, and possibly communicate with each other, a mechanism is needed to resolve resource conflicts. The third research axis, origins of cooperation, refers to how cooperative behavior is actually motivated and achieved. Here, we do not discuss instances where cooperation has been “explicitly engineered” into the robots’ behavior since this is the default approach. Instead, we are more interested in biological parallels (e.g., to social insect behavior), game-theoretic justifications for cooperation, and concepts of emergence. Because adaptability and flexibility are essential traits in a task-solving group of robots, we view learning as a fourth key to achieving cooperative behavior. One important mechanism in generating cooperation, namely,task decomposition and allocation, is not considered a research axis since (i) very few works in cooperative robotics have centered on task decomposition and allocation (with the notable exceptions of [126], [106], [134]), (ii) cooperative robot tasks (foraging, box-pushing) in the literature are simple enough that decomposition and allocation are not required in the solution, and (iii) the use of decomposition and allocation depends almost entirely on the group architectures(e.g. whether it is centralized or decentralized).Note that there is also a related, geometric problem of optimizing the allocation of tasks spatially. This has been recently studied in the context of the division of the search of a work area by multiple robots [97]. Whereas the first four axes are related to the generation of cooperative behavior, our fifth and final axis –geometric problems–covers research issues that are tied to the embed-ding of robot tasks in a two- or three-dimensional world. These issues include multi-agent path planning, moving to formation, and pattern generation.2.1. Group ArchitectureThe architecture of a computing sys tem has been defined as “the part of the system that remains unchanged unless an external agent changes it”[165]. The group architecture of a cooperative robotic system provides the infrastructure upon which collective behaviors are implemented, and determines the capabilities and limitations of the system. We now briefly discuss some of the key architectural features of a group architecture for mobile robots: centralization/decentralization, differentiation, communications, and the ability to model other agents. We then describe several representative systems that have addressed these specific problems.Centralization/Decentralization The most fundamental decision that is made when defining a group architecture is whether the system is centralized or decentralized, and if it is decentralized, whether the system is hierarchical or distributed. Centralized architectures are characterized by a single control agent. Decentralized architectures lack such an agent. There are two types of decentralized architectures: distributed architectures in which all agents are equal with respect to control, and hierarchical architectures which are locally centralized. Currently, the dominant paradigm is the decentralized approach.The behavior of decentralized systems is of-ten described using such terms as “emergence” and “self-organization.” It is widely claimed that decentralized architectures (e.g., [24], [10], [152],[108]) have several inherent advantages over centralized architectures, including fault tolerance, natural exploitation of parallelism, reliability, and scalability. However, we are not aware of any published empirical or theoretical comparison that supports these claims directly. Such a comparison would be interesting, particularly in scenarios where the team of robots is relatively small(e.g., two robots pushing a box), and it is not clear whether the scaling properties of decentralization offset the coordinative advantage of centralized systems.In practice, many systems do not conform toa strict centralized/decentralized dichotomy, e.g., many largely decentralized architectures utilize “leader” agents. We are not aware of any in-stances of systems that are completely centralized, although there are some hybrid centralized/decentralized architectures wherein there is a central planner that exerts high-levelcontrol over mostly autonomous agents [126], [106], [3], [36].Differentiation We define a group of robots to be homogeneous if the capabilities of the individual robots are identical, and heterogeneous otherwise. In general, heterogeneity introduces complexity since task allocation becomes more difficult, and agents have a greater need to model other individuals in the group. [134] has introduced the concept of task coverage, which measures the ability of a given team member to achieve a given task. This parameter is an index of the demand for cooperation: when task coverage is high, tasks can be accomplished without much cooperation, but otherwise, cooperation is necessary. Task coverage is maximal in homogeneous groups, and decreases as groups become more heterogeneous (i.e., in the limit only one agent in the group can perform any given task).The literature is currently dominated by works that assume homogeneous groups of robots. How-ever, some notable architectures can handle het-erogeneity, e.g., ACTRESS and ALLIANCE (see Section 2.1 below). In heterogeneous groups, task allocation may be determined by individual capabilities, but in homogeneous systems, agents may need to differentiate into distinct roles that are either known at design-time, or arise dynamically at run-time.Communication Structures The communication structure of a group determines the possible modes of inter-agent interaction. We characterize three major types of interactions that can be sup-ported. ([50] proposes a more detailed taxonomy of communication structures). Interaction via environmentThe simplest, most limited type of interaction occurs when the environment itself is the communication medium (in effect, a shared memory),and there is no explicit communication or interaction between agents. This modality has also been called “cooperation without communication” by some researchers. Systems that depend on this form of interaction include [67], [24], [10], [151],[159], [160], [147].Interaction via sensing Corresponding to arms-length relationships inorganization theory [75], interaction via sensing refers to local interactions that occur between agents as a result of agents sensing one another, but without explicit communication. This type of interaction requires the ability of agents to distinguish between other agents in the group and other objects in the environment, which is called “kin recognition” in some literatures [108]. Interaction via sensing is indispensable for modeling of other agents (see Section 2.1.4 below). Because of hard-ware limitations, interaction via sensing has often been emulated using radio or infrared communications. However, several recent works attempt to implement true interaction via sensing, based on vision [95], [96], [154]. Collective behaviors that can use this kind of interaction include flocking and pattern formation (keeping in formation with nearest neighbors).Interaction via communicationsThe third form of interaction involves explicit communication with other agents, by either directed or broadcast intentional messages (i.e. the recipient(s) of the message may be either known or unknown). Because architectures that enable this form of communication are similar tocommunication networks, many standard issues from the field of networks arise, including the design of network topologies and communications protocols. For ex-ample, in [168] a media access protocol (similar to that of Ethernet) is used for inter-robot communication. In [78], robots with limited communication range communicate to each other using the “hello-call” protocol, by which they establish “chains” in order to extend their effective communication ranges. [61] describes methods for communicating to many (“zillions”) robots, including a variety of schemes ranging from broadcast channels (where a message is sent to all other robots in the system) to modulated retroreflection (where a master sends out a laser signal to slaves and interprets the response by the nature of the re-flection). [174] describes and simulates a wireless SMA/CD ( Carrier Sense Multiple Access with Collision Detection ) protocol for the distributed robotic systems.There are also communication mechanisms designed specially for multiple-robot systems. For example, [171] proposes the “sign-board” as a communication mechanism for distributed robotic systems. [7] gives a communication protocol modeled after diffusion, wherein local communication similar to chemical communication mechanisms in animals is used. The communication is engineered to decay away at a preset rate. Similar communications mechanisms are studied in [102], [49], [67].Additional work on communication can be found in [185], which analyzes optimal group sizes for local communications and communication delays. In a related vein, [186], [187] analyzes optimal local communication ranges in broadcast communication.Modeling of Other Agents Modeling the intentions, beliefs, actions, capabilities, and states of other agents can lead to more effective cooperation between robots. Communications requirements can also be lowered if each agent has the capability to model other agents. Note that the modeling of other agents entails more than implicit communication via the environment or perception: modeling requires that the modeler has some representation of another agent, and that this representation can be used to make inferences about the actions of the other agent.In cooperative robotics, agent modeling has been explored most extensively in the context of manipulating a large object. Many solutions have exploited the fact that the object can serve as a common medium by which the agents can model each other.The second of two box-pushing protocols in[45] can achieve “cooperation without commun ication” since the object being manipulated also functions as a “communication channel” that is shared by the robot agents; other works capitalize on the same concept to derive distributed control laws which rely only on local measures of force, torque, orientation, or distance, i.e., no explicit communication is necessary (cf. [153] [73]).In a two-robot bar carrying task, Fukuda and Sekiyama’s agents [60] each uses a probabilistic model of the other agent. When a risk threshold is exceeded, an agent communicates with its partner to maintain coordination. In [43], [44], the theory of information invariants is used to show that extra hardware capabilities can be added in order to infer the actions of the other agent, thus reducing communication requirements. This is in contrast to [147], where the robots achieve box pushing but are not aware of each other at all. For a more com-plex task involving the placement of five desks in[154], a homogeneous group of four robots share a ceiling camera to get positional information, but do not communicate with each other. Each robot relies on modeling of otheragents to detect conflicts of paths and placements of desks, and to change plans accordingly.Representative Architectures All systems implement some group architecture. We now de-scribe several particularly well-defined representative architectures, along with works done within each of their frameworks. It is interesting to note that these architectures encompass the entire spectrum from traditional AI to highly decentralized approaches.CEBOTCEBOT (Cellular roBOTics System) is a decentralized, hierarchical architecture inspired by the cellular organization of biological entities (cf.[59] [57], [162] [161] [56]). The system is dynamically reconfigurable in tha t basic autonomous “cells” (robots), which can be physically coupled to other cells, dynamically reconfigure their structure to an “optimal” configuration in response to changing environments. In the CEBOT hierarchy there are “master cells” that coordinate subtasks and communicate with other master cells. A solution to the problem of electing these master cells was discussed in [164]. Formation of structured cellular modules from a population of initially separated cells was studied in [162]. Communications requirements have been studied extensively with respect to the CEBOT architecture, and various methods have been proposed that seek to reduce communication requirements by making individual cells more intelligent (e.g., enabling them to model the behavior of other cells). [60] studies the problem of modeling the behavior of other cells, while [85], [86] present a control method that calculates the goal of a cell based on its previous goal and on its master’s goal. [58] gives a means of estimating the amount of information exchanged be-tween cells, and [163] gives a heuristic for finding master cells for a binary communication tree. Anew behavior selection mechanism is introduced in [34], based on two matrices, the priority matrix and the interest relation matrix, with a learning algorithm used to adjust the priority matrix. Recently, a Micro Autonomous Robotic System(MARS) has been built consisting of robots of 20cubic mm and equipped with infrared communications [121].ACTRESSThe ACTRESS (ACTor-based Robot and Equipments Synthetic System) project [16], [80],[15] is inspired by the Universal Modular AC-TOR Formalism [76]. In the ACTRESS system,“robotors”, including 3 robots and 3 workstations(one as interface to human operator, one as im-age processor and one as global environment man-ager), form a heterogeneous group trying to per-form tasks such as object pushing [14] that cannot be accomplished by any of the individual robotors alone [79], [156]. Communication protocols at different abstraction levels [115] provide a means upon which “group cast” and negotiation mechanisms based on Contract Net [150] and multistage negotiation protocols are built [18]. Various is-sues are studied, such as efficient communications between robots and environment managers [17],collision avoidance [19].SWARM。
The PCP theorem by gap amplification
Hebrew University. Email: dinuri@cs.huji.ac.il. Supported by the Israel Science Foundation.
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• If x ∈ L then for any proof π , Pr[V π (x) accepts] ≤ 1 2. The PCP theorem says that every language in NP has a verifier that uses at most O(log n) random bits and reads only O(1) bits from the proof. In other words, Theorem 1.1 (PCP Theorem, [3, 2]) NP ⊆ PCP[log n, 1]. This theorem was a great surprise, as it completely revises our concept of a proof. Rather than the classical notion of a proof as a sequential object that if erroneous in even one place can easily prove a false statement. The PCP theorem provides a new proof notion that is more robust, and must be erroneous in many places when attempting to prove a falsity. Historically, the class P CP [r, q ] stemmed out of the celebrated notion of interactive proofs [20, 4] and the class IP. The original motivation for defining IP was cryptographic, but it soon lead to a list of remarkable complexity-theoretic results, including for example IP=PSPACE (see [24, 32]). We will not give a detailed historic account which can be found in, say, [1]. Let us just mention that an exciting sequence of papers (see [6, 14, 5]) lead to the following theorem: the class of all languages with exponential-sized proofs is equal to the class of all languages that can be verified by a (randomized) polynomial-time verifier. At this point attempts were made to “scale down” this result so as to characterize the class NP in similar terms, through suitable restriction of the verifier. This was especially motivated by the discovery of [12] that connected such a scale-down to an inapproximability result for the clique number (see below). This scale-down was achieved partially in [3] and completed in [2] and came to be known as the PCP theorem. The techniques that lead to the proof were mainly algebraic, including low-degree extension over finite fields, low-degree test, parallelization through curves, a sum-check protocol, and the Hadamard and quadratic functions encodings.
Obstructions to nonnegative curvature and rational homotopy theory
Obstructions to nonnegative curvature and rational homotopy theory
Igor Belegradek Vitali Kapovitch
Abstract We establish a link between rational homotopy theory and the problem which vector bundles admit complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if C lies in the class and T is a torus of positive dimension, then “most” vector bundles over C × T admit no complete nonnegatively curved metrics.
2000 Mathematics Subject classification. Primary 53C20, 55P62. Keywords: nonnegative curvature, soul, derivation, Halperin’s conjecture.
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Байду номын сангаас
Theorem 3.1). Then one is faced with the purely topological problem of recognizing whether a given vector bundle over C × T has this property. In other words, one needs to study the orbit of ξC × T under the action of the diffeomorphism group of C × T . Since vector bundles are rationally classified by the Euler and Pontrjagin classes, the problem reduces to analyzing the action of Diffeo(C × T ) on the rational cohomology algebra H ∗ (C × T, Q) of C × T . The “Taylor expansion” in T -coordinates of any self-diffeomorphism of C × T gives rise to a negative degree derivation of H ∗ (C, Q). One of the main points of this paper is that the orbit of ξC × T consists of bundles of the same form, unless there exists a negative degree derivation of H ∗ (C, Q) that does not vanish on the Euler or Pontrjagin classes of ξC . In particular, if H ∗ (C, Q) has no nonzero negative degree derivations, the above topological problem gets solved, which immediately implies that “most” bundles over C × T admit no complete metric of sec ≥ 0. To state our main results we need the following technical definition. Given a vector bundle ξ over C × T , we say that ξ virtually comes from C if for some finite cover p : T → T , the pullback of ξ by idC × p is isomorphic to the product ξC × T where ξC is a bundle over C . If ξ virtually comes from C , then no known method can rule out the existence of a complete metric with sec ≥ 0 on the total space E (ξ ) of ξ , and potentially all such bundles might be nonnegatively curved. In this paper we show that the converse is often true, namely, under various assumptions on C , we show that, if ξ is a vector bundle over C × T such that E (ξ ) admits a complete metric with sec ≥ 0, then ξ virtually comes from C . This happens for any C if ξ has rank two. Theorem 1.1. Let C be a closed smooth simply-connected manifold, and T be a torus. Let ξ be a rank two vector bundle over C × T . If E (ξ ) admits a complete metric with sec ≥ 0 , then ξ virtually comes from C . Oriented R2 -bundles over C × T are in one-to-one correspondence via the Euler class with H 2 (C × T, Z) ∼ = H 2 (C, Z) ⊕ H 2 (T, Z). Thus, any oriented R2 -bundle ξ over C × T can be written uniquely as cξ + tξ where cξ ∈ H 2 (C, Z), tξ ∈ H 2 (T, Z). Theorem 1.1 implies that if sec(E (ξ )) ≥ 0, then tξ = 0. More generally, it is easy to see that “most” vector bundles over C × T do not virtually come from C , at least when dim(T ) is large enough (for a precise result, see [BK01b, 4.4, 4.6] and Lemma B.1 below). In fact, ξ virtually comes from C iff all rational characteristic classes of ξ lie in the H ∗ (C, Q)⊗H 0 (T, Q)-term of the K¨ unneth decomposition ⊕i H ∗ (C, Q)⊗H i (T, Q) ∗ of H (C × T, Q). One of the main sources of examples of closed manifolds of nonnegative curvature is given by homogeneous spaces or, more generally, biquotients of compact Lie groups. In this case we prove Theorem 1.2. Let C = G//H be a simply connected biquotient of compact Lie groups such that H is semi-simple, and let T be a torus. Let ξ be a vector bundle over C × T of rank ≤ 4 . If E (ξ ) admits a complete metric with sec ≥ 0 , then ξ virtually comes from C . 2
no-gap序列合成模式
no-gap序列合成模式English Answer:No-Gap Sequencing Mode.The no-gap sequencing mode is a sequencing mode in which the sequencing process is performed continuously, without stopping between reads. This is in contrast to the gap sequencing mode, in which the sequencing process is paused between reads to allow for the insertion of a gap between the reads.The no-gap sequencing mode offers several advantages over the gap sequencing mode. First, it is faster. By eliminating the need to pause between reads, the no-gap sequencing mode can significantly reduce the sequencing time. Second, it is more accurate. By avoiding the introduction of gaps between reads, the no-gap sequencing mode can reduce the risk of errors in the sequencing process. Third, it is more efficient. By eliminating theneed for gap filling, the no-gap sequencing mode can reduce the cost and time required to complete the sequencing process.However, the no-gap sequencing mode also has some disadvantages. First, it is more difficult to perform. The continuous sequencing process requires more precise control of the sequencing process, which can be difficult to achieve. Second, it is more likely to generate sequencing errors. The continuous sequencing process can lead to the accumulation of errors, which can make it difficult to obtain accurate sequencing data.Overall, the no-gap sequencing mode offers several advantages over the gap sequencing mode. However, it is also more difficult to perform and more likely to generate sequencing errors. The choice of which sequencing mode to use will depend on the specific application.Chinese Answer:无间隙测序模式。
Morita theory of comodules
Introduction There is a long tradition of using Morita theory in the study of Hopf-Galois extensions and by generalization Galois corings and Galois comodules, see e.g. [15], [16], [1], [10], [4]. One of the applications of Galois theory within the context of corings and comodules is corresponding (generalized) descent theory. That is, a study of the adjoint Hom and tensor functors between the category of comodules over a coring and the category of modules over an appropriately chosen algebra. In particular to find sufficient and necessary conditions for these functors to be full and faithful. Results of this kind are referred to as (weak and strong) structure theorems. If a coring C is finitely generated and projective as left A-module, then its category of right comodules becomes isomorphic to the category of modules over the dual ring ∗C . Hence Galois theory for such a coring describes in fact functors between two module categories, which explains the relation with Morita theory. Although in general Morita contexts for comodules can be constructed without any finiteness restriction on the coring C , strictness of these Morita contexts implies such a finiteness condition for C , and usually as well for the comodule Σ, appearing in the Morita context (see [4, Lemma 2.5]). The standard result in Morita theory says that the connecting maps ▽ and of a Morita context (A, A′ , P, Q, ▽, ) are bijective (or equivalently surjective, if the algebras A and A′ have a unit) if and only if the Morita context induces an equivalence of the module categories MA and MA′ . It was a natural question to pose how distinct any Morita context is from an equivalence of categories. This question has in fact two categorically dual answers which were found by several authors. They say that any Morita context induces an equivalence between certain quotient categories of the original module categories (see [24]) as well as an equivalence between certain full
Gaps in √ n mod 1 and ergodic theory Contents
2
We emphasize that the function F (t) is not analytic or even C 3 at the points t = 1/2 and t = 2. The gap distribution has genuine phase transitions at these two critical points. Moreover, the tail of the distribution is not exponential; instead, we have F (t) ∼ (3/π 2 )t−3 as t → ∞. Thus large gaps √ are much more likely for { n }N 1 than for N random points (although both are rare events). √ From gaps to lattices. The distribution of gaps in { n } is related, via ergodic theory, to the probability p(t) that a random lattice translate Λ ⊂ R2 meets a given triangle St of area t. To explain this relation, we first discuss spaces of lattices, their natural measures and the dynamical systems they support. Recall that a lattice Λ0 ⊂ R2 is a discrete subgroup isomorphic to Z2 ; it is unimodular if the quotient torus R2 /Λ0 has area one. A lattice translate is simply a coset Λ = v + Λ0 ⊂ R2 . The space of all translates of unimodular lattices in R2 can be naturally identified with the homogeneous space E = ASL2 (R)/ ASL2 (Z). (1.2)
Convexity of chance constraints with dependent random
(i = 1, 2).
Then, for p∗ ≈ 0.7 one has that M (p) is nonconvex for p < p∗ whereas it is convex for p ≥ p∗ . Note, that though the multivariate normal distribution is log-concave (see [Pre95]), the result by Pr´ ekopa mentioned above does not apply because the gi are not concave. As a consequence, M (p) is not convex for all p as it would be guaranteed by that result. Nonetheless, eventually convexity can be verified by the tools developed in [HS08]. The aim of this paper is to go beyond the restrictive independence assumption made in [HS08]. While to do this directly for arbitrary multivariate distributions of ξ seems to be very difficult, we shall see that positive results can be obtained in case that this distribution is modeled by means of a copula. Copulae allow to represent dependencies in multivariate distributions in a more efficient way than correlation does. They may provide a good approximation to the true underlying distribution just on the basis of its one-dimensional margins. This offers a new perspective also to modeling chance constraints not considered extensively so far to the best of our knowledge. The paper is organized as follows: in a first section, some basics on copulae are presented and the concepts of r-concave and r-decreasing functions introduced. The following section contains our main result on eventual convexity of chance constraints defined by copulae. In a further section, log-exp concavity of copulae, a decisive property in the mentioned convexity result, is discussed. Finally,
An Isoperimetric Inequality With Applications To Curve Shortening
2L2A
’71"
>2
fv p
ds.
Adding these two inequalities yields
LA
(L + L)A
> (. pds
which proves inequality A. Inequality B follows from lemma 1.
Remark. H. Jacobowitz has observed that the bone shaped nonconvex curve in Figure shows that inequality 13 does not hold for all simple closed curves. The curve is made of straight lines and circular arcs with curvature 1/R. The quantity xds is approximately 2r/R. If the shank of the bone is long enough so that the area and length of the ends can be neglected, then the ratio L/A 1/e. When e is small enough inequality B fails and an attempt to shorten such a curve by the process discussed in the next section leads to an initial increase in the isoperimetric ratio.
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Gap Theorem on Complete Noncompact Riemannian Manifold
r
sk (x0 , s)ds = o(log r)
0
(*)
1 where k (x0 , s) = vol(B ( ahler manifold. x0 ,s)) B (x0 ,s) R(x)dv . They got the same result on K¨ Stimulated by Ni and Tam’s result, we consider condition (∗) on the locally conformally flat Riemannian manifold, and also get the analogous gap theorem.
Cheng Bing ZHAO1,2
1. Department of Mathematics, Anhui University of Architecture, Anhui 230022, P. R. China; 2. Postdoctoral Research Station of Management College, Hefei University of Technology, Anhui 230009, P. R. China Abstract A gap theorem on complete noncompact n-dimensional locally conformally flat Riemannian manifold with nonnegative and bounded Ricci curvature is proved. If there holds the following condition:
Received March 26, 2009; Accepted July 3, 2009 Supported by the National Natural Science Foundation of China (Grant No. 70631003), the Natural Science Foundation of Anhui Education Department (Grant No. KJ2011A061), the Natural Science Foundation of Anhui Science and Technology Department (Grant No. 1104606M01) and the Doctor of Philosophy Foundation of Anhui University of Architecture (Grant No. 2007-6-3). E-mail address: chengbingzhao@
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Proposition 2.1 Let M be an n-dimensional (n ≥ 3) complete noncompact Riemannian manifolds with nonnegative Ricci curvature. If the scalar curvature is bounded, then the Yamabe
r
sk(x0 , s)ds = o(log r )
0
then the manifold is flat. Keywords Ricci curvature; conformally flat; gap theorem. Document code A MR(2010) Subject Classification 53C20; 53C21 Chinese Library Classification O186.1
Gap theorem on complete noncompact Riemannian manifold
431
flow (2.1) has a smooth solution on a maximal time interval [0, tmax ) with tmax > 0 such that either tmax = +∞, or the evolving metric contracts to a point at a finite time tmax in the sense that for any curve γ on M, the length of γ with respect to the evolving metric gij (x, t) tends to zero as t → tmax . Now we want to drive some estimate for the solution of the Yamabe flow (2.2). From above proposition we have a smooth solution gij (x, t) of (2.1) on a maximal time interval [0, tmax ). 4 Write the solution as gij (x, t) = (u(x, t)) n−2 gij (x), where u(x, t) is a positive solution of (2.2) on [0, tmax ). Lemma 2.1 For r → +∞, t > 0, we have −
1. Introduction
Let M be an n-dimensional complete noncompact Riemannian manifold with nonnegative Ricci curvature. Mok, Siu and Yau [7] proved that if a complete noncompact K¨ ahler-Stein manifold of nonnegative and bounded holomorphic bisectional curvature of complex dimension n ≥ 2 has maximal volume growth and the scalar curvature decays faster than quadratic, in the sense that, for some C > 0 and ε > 0, R(x) ≤ Cd(x0 , x)−(2+ε) , then M is isometrically biholomorphic to C n . This can be interpreted as a gap phenomenon of the bisectional curvature on K¨ ahler manifolds (A more general theorem in Riemannian category was proved by Greene and Wu in [10]). Later the similar result was extended to the Riemannian manifold with maximal volume growth and nonnegative Ricci curvature by Bando, Kasue and Nakajima in [11]. Recently, Chen and Zhu obtained the gap theorem on the locally conformally flat manifolds [2]. They showed that: Let M be an n-dimensional (n ≥ 3) complete noncompact locally conformally flat Riemannian manifolds with nonnegative Ricci curvature. If the scalar curvature is bounded and there exists
n+2 where N = n −2 , △ is the Laplace operator with respect to the initial metric gij (x). We know from [2] that the Yamabe flow (2.1) has a smooth solution, namely,
430
C. B. ZHAO
a positive function ε : R → R with limr→∞ ε(r) = 0 such that 1 vol(B (x0 , r))
B (x 0 , r )
R(x)dv ≤
ε(r) , for x0 ∈ M, r > 0, r2
(1)
then M is flat. Later in [3], they used the theory of the Ricci flow to obtain the analogous gap theorem on K¨ ahler manifold as follows. Suppose M is a complete noncompact K¨ ahler manifold of complex dimension n ≥ 2 with bounded and nonnegative holomorphic bisectional curvature, and the condition (1) is satisfied, then M is flat. In [9], Ni and Tam changed condition (1) into the following condition (∗)