in a news distribution system with a tremendous number of users , such as digg.com and reddit.com , thousands of news are released every hour .when a user logs on the system without any support , he is drowned in the flood of information immediately  .recommender systems emerged to help users deal with the information overload and find out content matching their interests  . in traditional recommender systemsrelevant news are delivered to users according to their profiles and historical activities , with recommendation mechanism designed in a centralized way : data analysis and recommendation decision are made by the system for the whole network . in contrast , in social recommender systems users submit and forward news to others , as in a peer - to - peer network .the core idea of peer - to - peer is that all nodes in the system have the same function : each node can both gain service from other nodes , as a client , and provide service to others , as a server .  in this work we present a modified version of a previously introduced news recommender system  .this system is naturally implemented in a peer - to - peer environment : each user , considered as a node of the network , does not get news directly from some central servers , but he can post news and recommend them to other users ( i.e. his followers ) and at the same time he receives news from other users ( i.e. his leaders ) .hence each node acts as a follower and at the same time can become a leader in the network .when a user receives a news from his leader and likes it , he forwards it to his followers ; then , if these followers like the news too , they forward it further to their followers , and so on . hence news diffusion over the network resembles an epidemic spreading  .as in peer - to - peer systems , the leader - follower network is varying in topology : links between leaders and followers may be changed as the network evolves .the system continually refines the neighborhood of each user until connections between taste mates are established , so that news can spread quickly from the original source to the users who are most interested in it .network evolution is therefore driven from an interplay between topology and dynamics  .the leader updating procedure is a crucial part of the whole recommender mechanism .an effective updating method should choose for each user a leader with very similar tastes at very low cost .in this work we compare several heuristic methods in detail .we discuss their efficiency , performance and scalability ( which are important aspects for real systems ) , and we analyze the properties of the evolving network structure with agent - based simulations .a comparison with other widely adopted popularity - based recommendation methods has already been discussed in  .this work is organized as follows . in section 2we present related work . in section 3we describe the news recommendation method and an agent - based model to test it . in section 4we report simulation results for network s evolution with different updating methods . in section 5we introduce two refined updating methods .lastly , we conclude with discussion .so far , plenty of recommendation techniques have been proposed by researchers .these techniques can be classified in three main categories : content - based , collaborative filtering and social filtering methods .content - based methods  try to classify items according to their content ( e.g. by keywords ) and to extract users preferences from their pick history .these two elements are used to obtain recommendation : items recommend to a user are the ones whose characteristics match user s preferences .collaborative filtering methods make predictions about the interests of a user by collecting taste information from the community the user belongs to , with the underlying assumption that those users who agreed in the past tend to agree again in the future . in memory - based collaborative filtering , such as in amazon.com  ,the recommendation score for an item is computed as a weighted average of ratings given by other users with weights proportional to the similarity between users  . in model - based collaborative filtering methods ( e.g. bayesian clustering  and plsi  )users are classified according to their preferences .diffusion - based methods , where heat conduction  , random walk  or hybrid processes  are employed to efficiently extract the taste similarity between users or items , can be considered as extensions of collaborative filtering .social filtering methods make use of users relationship in a social network to obtain recommendations .these methods are based on the assumption that users prefer recommendations made by a friend than those by a centralized system  . in other words ,social influence is more determinant for users to decide what to read than similarity of past activities  .generally , a social recommender system can be devised in two different ways : pull and push . in the first case ,each user selects other users as his information sources and he can pull information from these sources ( as in , e.g. , delicious.com ) . in the latter ,each user can push what he likes to other users who have accepted him as an information source ( as in , e.g. , twitter.com ) . in many of these systemsusers have also the possibility to assign information with user - defined words , so - called _ tags _ , that allow a more effective management of resources by users  .different from all methods mentioned above , the news recommendation mechanism we study in this work combines memory - based and social recommendation together with network s topology optimization .recommendation scores are computed using users past ratings , though recommendations come only from the information sources a user selects ; in parallel , the leader - follower network is continually refined according to similarity of user tastes , in order for news to be efficiently delivered to their eagerest readers .this combination of social and similarity recommendation turns out to be particularly effective : the presence of directed links among users makes the recommendation process naturally fast and scalable , while the use of rating patterns similarity in the information diffusion process enhance users social contacts and experience .in this section we describe the news recommendation mechanism  that provides a personalized set of news for each user .as stated in the introduction , the system consists of a set of users with distinct tastes who continually post and evaluate various news items .   the taste similarity between usersis predicted using statistical correlation of users rating histories .we adopt pearson correlation coefficient : for a pair of users and  where and are the ratings given by these users to news and and are the average ratings of and . ] .hence the starting value of is equal to .as the network evolves , the reciprocity coefficient grows from with all updating methods : network s structure becomes more and more reciprocal .this growth eventually stops when a stationary state of the network is found .if the network is driven to the ground state ( e.g. by the selective method ) the value of the reciprocity becomes =2/(d+1) ] ( i.e. the probability to find a link between two users and who are linked to user and who are taste - mates of user ) . , like , corresponds to users randomly choosing leaders among their best taste - mates .instead with the brl method , where links can only form locally , the clustering coefficient becomes immediately very large ( even bigger than ) : network s evolution is hindered because a high clustering coefficient means that news do not spread effectively ( they get few directions to propagate and their traffic is limited to a subset of the network ) .users who happen to be far apart are unlikely to read the same news : their similarity remains undiscovered and they can not be connected even if they are taste - mates .note also that with this method once a node has no followers , it will never get any follower in future because it can not be selected as a leader , hence some users may get isolated from the system .these two factors cause the network evolution to end in a sub - optimal equilibrium state .in this section we propose new updating methods that strongly enhance the system s performance .the brl method has the ability to exploit the local neighborhood network , that is the most significant feature of a peer - to - peer system .unfortunately , as we have mentioned , it has some major drawbacks caused by the limited range of the network that this method is able to explore .we attempt to overcome these problems by broadening the exploration range .the first possible solution is to explore the whole second - order neighborhood ( i.e. , not only the subset related to the best leader ) , or even to go further by including the third - order neighborhood , fourth - order neighborhood , and so on .however , the computational cost of such exploration grows very fast it is for a -order neighborhood and soon becomes comparable with the cost of the gmb method .  )compared to other leader updating methods .inset : comparison with gmb for a smaller system ( ) . ]a more effective solution is to employ some randomness in the updating method in order to have the possibility to connect users regardless of their distance and prevent the system from getting trapped in a sub - optimal state .hence we propose a _ hybrid _updating method that combines selective and brl methods : at each updating step we employ the first with probability , otherwise we employ the second . this mechanism mimics the evolution of self - organizing communities where users search for friends among friends of friends but casual encounters also occur .figure [ fig7 ] shows that the hybrid method improves system s performance as compared to both selective and brl methods , still keeping an affordable computational cost .hence it represents a cheap and effective updating algorithm suitable for large systems .we now concentrate on the drawbacks of the above mentioned random updating methods .the pure random method may replace an old leader with a less similar one , hence it can not reach the best assignment of leaders . therefore the selective method outperforms the pure random .however the selective method may leave the network unchanged if it does nt find a better leader ( which may occur quite often ) , hence slowing down the system s evolution .a simple solution to this problem is to employ a selective approach with more trials .we name this new method _las vegas algorithm_  .its basic idea is that if the newly randomly selected candidate leader is less similar than the old leader , we drop it and retry the random updating until a better candidate is found .since there may not be a better candidate , we set the maximum number of trials to a predefined value .the bigger this value , the longer the execution time , but also the higher chances to find a better leader .compared to other updating methods .inset : comparison with gmb for a smaller system ( ) . ]figure [ fig8 ] shows the performance of the las vegas algorithm for different values of .note that the selective method is nothing but a las vegas method with .as increases from 1 , the convergence rate of the las vegas method becomes faster and faster , and it becomes even comparable with the gmb method , yet keeping the affordable computational cost of gmb .moreover , the average number of actual trials of the las vegas algorithm ( figure [ fig9 ] ) is very small at the beginning of system s evolution , when it is very easy to find better candidate leaders for users : again reduces computational cost. then the network approaches to the ground state and as each user is already linked to his taste mates ; however at this stage the evolution already ended .las vegas algorithm therefore improves both effectiveness and efficiency of the leader updating schedule , standing as a candidate mechanism to be employed in real recommender systems .   of the las vegas algorithm with at different stages of the system s evolution ( represented by the value of the excess differences ) . at the beginning , ( the fictitious starting point at only represent the initial network s state ) . as the system approaches the ground state , ( represents the point of convergence ) . ]online news websites are nowadays overloaded with a variety of fresh information .the hard task for readers is how to find the news they are really interested in .recommender systems are a possible answer to this problem .a significant yet less noticed aspect is that social relationship is extremely useful in delivering news to potentially interested users .  in this work we generalized a recently proposed news recommendation method  to make it more efficient and effective .this method combines memory - based and social recommendation : recommendation scores of news items are computed using users past ratings , though recommendations come only from the information sources a user selects .another important feature of this model is that network s topology can be continually refined according to similarity of user tastes , in order for news to be efficiently delivered to the users that are most interested in them .we proposed new stochastic algorithms which are computationally cheap ( and hence scalable with system s size ) and provide an effective way to speed up the leader updating process .the next step will be to develop the described system on a real platform .there are still some open questions to be answered .for instance , users do not differ only in their particular tastes but in many other aspects , like activity , reputation , generosity , and so on .news are also different in many aspects , like quality and freshness . taking into account users and news heterogeneitymay bring to a more realistic scenario .another important point is that any real recommender system must face potential attacks from malicious users or communities , that may systematically introduce spam news or try to intentionally mislead the system .measures of trustworthiness and reputation of users can be introduced to cope with these threats .we acknowledge helpful discussions with chiho yeung , cihang jin and matus medo .this work was partially supported by swiss national science foundation ( grant no .200020 - 121848 ) and by the future and emerging technologies programmes of the european commission fp7-ict-2007 ( project liquidpublication , grant no .213360 ) and fp7-cosi - ict ( project qlectives , grant no .t.z . acknowledges the national natural science foundation of china under grant nos .60973069 .g. m. d. corso , a. gulli , f. romani , proc .of the 14th intl . conf . on world wide web .new york , ny , usa : acm press ( 2005 ) 97 .r. burke , user modeling and user - adapted interaction , 12(4 ) ( 2002 ) 331 .b. sarwar , g. karypis , j. konstan , j. riedl , proc . of the 10th intl. conf . on world wide web .new york , ny , usa : acm press ( 2001 ) 285 .p. resnick , n. iakovou , m. sushak , p. bergstrom , j. riedl , proc . of computersupported cooperative work conf . , ( 1994 )m. medo , y .- c .zhang , t. zhou , europhys .88 ( 2009 ) 38005 .r. pastor - satorras , a. vespignani , phys .lett . , 86 ( 2001 ) 3200 .t. zhou , z. q. fu , b .- h .wang , prog ., 16 ( 2006 ) 452 .d. garlaschelli , a. capocci , g. caldarelli , nature physics 3 ( 2007 ) 813 .r. k. pon , a. f. cardenas , d. buttler , t. critchlow , ieee symposium on computational intelligence and data mining , honolulu , hi ( 2007 ) .i. cantador , p. castells , proc . of recsys09 workshop 3 : workshop on context - aware recommender systems ( 2009 ) .g. linden , b. smith , j. york , ieee internet computing , 7(1 ) ( 2003 ) 76 .j. breese , d. heckerman , c. kadie , proc . of the 14th conf . on uncertainty in artificial intelligence ( 1998 ) 43 .t. hofmann , acm trans . on information systems , 22(1 ) ( 2004 ) 89 .c . zhang , m. blattner , y .- k .yu , phys .99 ( 2007 ) 154301 .t. zhou , j. ren , m. medo , y .- c .e , 76 ( 2007 ) 046115 .t. zhou , z. kuscsik , j. g. liu , m. medo , j. r. wakeling , y .- c .zhang , proc ., 107 ( 2010 ) 4511 .r. sinha , k. swearingen , proc . of the delosnsf workshop on personalization and recommender systems in digital libraries , dublin , ireland ( 2001 ) 271 .s. p. borgatti , r. cross , manage .49(4 ) ( 2003 ) 432 .a. gulli , s. cataudella , l. foschini , proc . of the 6th intl .workshop on algorithms and models for the web - graph , barcelona , spain ( 2009 ) 143 .a. capocci , g. caldarelli , j. phys .a : math . theor .41 ( 2008 ) 224016 .k . zhang , c. liu , j. stat .( 2010 ) p10005 .k . zhang , c. liu , y .- c .zhang , t. zhou , epl 92 ( 2010 ) 28002 .m. blum , r. floyd , v. pratt , r. rivest , r. tarjan , j. comput .( 1973 ) 448 .a. singh , m. castro , p. druschel , a. rowstron , proc . of the acmsigops european workshop ( 2004 ) .s. wasserman , k. faust , social network analysis , cambridge university press , cambridge ( 1994 ) .g. fagiolo , phys . rev .e 76 ( 2007 ) 026107 l. babai , technical report dms 79 - 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Abstract: recommendation systems represent an important tool for news distribution on the internet . in this work we modify a recently proposed social recommendation model in order to deal with no explicit ratings of users on news . the model consists of a network of users which continually adapts in order to achieve an efficient news traffic . to optimize network s topology we propose different stochastic algorithms that are scalable with respect to the network s size . agent - based simulations reveal the features and the performance of these algorithms . to overcome the resultant drawbacks of each method we introduce two improved algorithms and show that they can optimize network s topology almost as fast and effectively as other not - scalable methods that make use of much more information .   recommender systems , social recommendation , adaptive networks


the chemostat was introduced in the fifties as an experimental device to study the microbial growth on a limiting resource .it is also often used as a mean to reproduce situations where ( limiting ) nutrients are fed to micro - organisms , typically in a liquid medium , in natural ecosystems or anthropized environments .more generally , the chemostat is largely used as a scientific investigation tool in microbial ecology .the mathematical model of the chemostat has been extensively studied ( see e.g. ) and used as a reference model in microbiology , microbial ecology or biotechnological industries such as the waste - water treatment . more generally , the chemostat serves to describe resource - consumer relations , where the resource is supplied at a constant rate .however , in many applications , the assumption of perfectly stirred chemostats is , in general , too restrictive . in the eighties , the gradostat , as an experimental device composed of a set of chemostats of identical volume interconnected in series ,was introduced to represent spatial gradient , in a marine environment or to model rhizosphere .it motivated several mathematical studies .similarly , an interest for series of bioreactors appeared in biochemical industry , with tanks of different volumes to be minimized . in ecology ,island models have been proposed since the late sixties to study the effects of heterogeneous environments with more general patterns than serial ones .several studies of prey - predator in patchy environments have been conducted since then .comparatively , relatively few studies have considered non - serial interconnections for resource - consumer models or chemostats . in natural reservoirs such as in undergrounds or ground - waters , a spatial structure with interconnections between several volumesis often considered , each of them being approximated as perfectly mixed tank .those interconnections can be parallel , series or built up in more complex networks . to our knowledge, the influence of the topology of a network of chemostats on the overall dynamics has been sparsely investigated in the literature .however , the simple consideration of two interconnected habitats can lead to non - intuitive behaviors and influence significantly the overall performances .recently , literature in ecology has raised the relevance of `` source - sink '' models for describing plants / nutrients interactions , and predicting ecosystems performances .those models are mathematically close to general gradostat models , but with a significant difference concerning the resources compartments , for which the input rate mechanisms ( due to atmospheric depositions or rock alterations ) are assumed to be independent of the nutrient leaching ( and not modeled as a transport term as in hydrology or in chemostat - like models ) .it is also well - known since the seventies that microbial growth can be inhibited by large concentrations of nutrient .such inhibition can be modeled by non - monotonic response functions and lead to initial - condition dependent washout .non - monotonic response functions occur in predator - prey models , for instance , when the predation decreases due to the ability of the prey to better defend when their population get larger .this non - monotonic functional response could also lead to bi - stability and possible extinction of the predator .several control strategies of the input flow were proposed in the literature to globally stabilize the chemostat but the ability of a spatial structure to passively stabilize such dynamics has not been yet studied ( in a general structure of networks of chemostats is considered but with monotonic growth rates , while in non - monotonic functions are considered but for the serial gradostat only ) .  the present work considers a particular interconnection of two chemostats of different volumes , one being a buffer tank . to our knowledge , this spatial structure , that is neither serial nor parallel , has not yet been considered in the literature . this structure is analogous to refuges in patchy environments , but here both consumer and resource are present in each vessel .we prove that it is possible with such a configuration to obtain repulsive washout equilibrium , while any serial , parallel or single tank structures with the same total volume exhibits multi - stability .this result brings new insights into the role of spatial patterns in the stability of bio - conversion processes in natural environments , where buffers can occur such as in soil ecosystems .it has also potential implications for the design of robust industrial bio - processes .the paper is organized as follows .section [ hypo ] presents the hypotheses and the buffered configuration , comparing with serial and parallel interconnections .section [ section3 ] studies the multiplicity of equilibria and their stability for such configurations , considering a general class of response functions ( monotonic as well as non - monotonic ) .section [ section4 ] discusses the biological and ecological implications of the results of section [ section3 ] in terms of persistence of microbial species in a non - homogeneous environment , along with some industrial perspectives .numerical simulations illustrate the results on an haldane function in section [ section_example ] .all the proofs are postponed to the appendix .we consider the chemostat model with a single strain growing on a single limiting nutrient . the system is fed with nutrient of concentration with flow rate .the total volume is assumed to be constant ( i.e. input and output flow rates are supposed to be identical ) .when the concentrations of nutrient ( or substrate ) and biomass , denoted respectively and , are homogeneous , as it is the case in perfectly mixed tanks , the system can be modeled by the well - known differential equations : where is the uptake function and the yield coefficient of the transformation of nutrient into biomass . without any loss of generality ,we take ( at the price of changing in ) . for convenience ,we define the dilution rate we consider quite general uptake functions , that fulfill the following properties .+ * assumptions a1 . ** the function is analytic and such that , for any .* the function is either increasing , or there exists such that is increasing on and decreasing on .the usual uptake functions , such as the monod function  or the haldane one  fulfill theses hypotheses .classically , we consider the set that plays an important role in the determination of the equilibria of the system . under assumptions a1 ,the set is either empty or an open interval that we denote where can be equal to .+ we recall from the theory of the chemostat model ( see for instance ) that under assumptions a1 there are three kinds of phase portrait of the dynamics ( [ chemostat0 ] ) , depending on the parameter .[ prop - chemostat ] assume that hypotheses a1 are fulfilled .  * _ case 1 : _ or .the washout equilibrium is the unique non negative equilibrium of system ( [ chemostat0 ] ) .furthermore it is globally attracting .* _ case 2 : _ .the system ( [ chemostat0 ] ) has three non - negative equilibria , and . only and are attracting , and the dynamics is bi - stable . * _ case 3 : _the system ( [ chemostat0 ] ) has two non negative equilibria and . is globally attracting on the positive quadrant .notice that in case 2 , the qualitative behavior of the growth can change radically depending on the initial condition .+ the question we investigate in this paper is related to the assumption that the vessel is perfectly mixed , and to the role that a spatial structure could have on the stability of the dynamics .consider the case for which the washout equilibrium is attracting in the chemostat model ( cases 1 and 2 of proposition [ prop - chemostat ] ) .furthermore , consider spatial configurations with the same input flow and residence time than the perfectly mixed case , i.e. with the same total volume and input flow .then , one has the following property .[ lemmaserialparallel ] assume that hypotheses a1 are fulfilled and let and be such that .then the washout is an attracting equilibrium in at least one vessel of any interconnection in series or in parallel of tanks of volume such that , assuming that each of them is perfectly mixed .this lemma shows that when a bacterial species can not persist in a chemostat , from any or a subset of initial conditions , this property persists in at least one vessel of any serial or parallel interconnection of chemostats with the same total volume . in the present work , we study a different kind of spatial configuration with an asymmetry created by two interconnected volumes , one of them serving as a buffer ( see figure [ fig - pocket ] ) .we call these spatial configurations a `` buffered chemostat '' , to be compared with the `` single chemostat ''.  , width=151 ]   and are respectively the volumes of the main tank and the buffer , and and denote the input flow rates of each tank , with .we assume that each vessel is perfectly mixed .straightforwardly , the dynamical equations of the buffered chemostat are notice that the limiting case consists in a by - pass of the volume with a flow .+ in the next section , we study the equilibria of this model , their multiplicity and their stability .given a volume and an input flow rate , we describe the set of all possible buffered configurations with and by two parameters and defined as follows this choice of parameterization is more convenient than the original one because it decouples more easily the role of the two parameters , as it is shown by equations ( [ chemostat2b ] ) below.  dynamics ( [ chemostat2 ] ) can then be written in the following way \dot x_{1 } & = & \ds \mu(s_{1})x_{1}+d\frac{\alpha(1-r)(x_{2}-x_{1})- ( 1-\alpha(1-r))x_{1}}{r } \ , \\[2 mm ] \dot s_{2 } & = & \ds -\mu(s_{2})x_{2}+d\alpha(s_{in}-s_{2 } ) \, \\[2 mm ] \dot x_{2 } & = & \ds \mu(s_{2})x_{2}-d\alpha x_{2 } \ . \end{array}\ ] ] at equilibrium , one should have that is .then , one should have that is . thus, equilibria of dynamics ( [ chemostat2b ] ) can be written as solutions of the following equations : due to the cascade structure of model ( [ chemostat2 ] ) , the study of the dynamics of the second reactor can be done independently of the first one .depending of the value of , the three cases given in proposition [ prop - chemostat ] for the single chemostat are possible in the second tank .this implies the following two possibilities for the equilibria of the first sub - system .  1 .when converges to the washout equilibrium ( cases 1 and 2 ) , the dynamics is asymptotically equivalent to a single chemostat model with dilution rate , and proposition [ prop - chemostat ] applies .when converges towards a positive equilibrium ( cases 2 and 3 ) , we consider the family of hyperbola that are the graphs of the functions parameterized by and . from equations ( [ eq1 ] ) and ( [ eq2 ] ) , a positive equilibrium of ( [ chemostat2b ] ) satisfies or equivalently is the abscissa of an intersection of the graph of with the hyperbola .then , from equation ( [ eq2 ] ) , to each solution corresponds a unique .notice that the washout is not an equilibrium for the first tank .  in the following ,we consider only non - trivial cases for which the second tank admits a positive equilibrium , assuming the hypotheses : + * assumptions a2 .* under assumptions a1 , and are positive numbers such that and .+ similar to the single chemostat that considers the set given in ( [ lambda ] ) , we define the set   we shall also consider the subset of configurations for which system ( [ chemostat2b ] ) admits an unique positive equilibrium , denoted by   we state now our main results .  [ main_theo ] assume that hypotheses a1 and a2 are fulfilled .the set is non - empty , and for almost any one has the following properties , except from a subset of initial conditions of zero lebesgue measure .* when the initial condition of the sub - system belongs to the attraction basin of , the solution of system ( [ chemostat2b ] ) converges exponentially to the rest point when , or to the washout equilibrium when .* when the initial condition of the sub - system does not belong to the attraction basin of , the trajectory of the system ( [ chemostat2b ] ) converges exponentially to a positive equilibrium + where is the left endpoint of a connected component of .  moreover , the set is non - empty .let give some observations on these results .* in contrast to the single chemostat , for which the set could be empty , the set is non - empty .this means that dynamics ( [ chemostat2b ] ) always admits a positive equilibrium , even when the washout is the only equilibrium of the single chemostat , contrary to serial or parallel chemostats ( cf lemma [ lemmaserialparallel ] ) .* when the initial condition of the sub - system belongs to the attraction basin of ( that could be reduced to a singleton ) , it is a not a surprise that the asymptotic behavior of the sub - system is the same as for a single chemostat with a dilution rate equal to ( cf point i. ) .otherwise , the whole state converges to a positive equilibrium , with a possible multiplicity of equilibria ( cf point ii . ) . here, a remarkable feature is the existence of buffered configurations that possess an unique globally asymptotically stable equilibrium ( when and ) , in contrast to the single chemostat or any serial or parallel configurations for which a bi - stability occur when the functional response is non - monotonic .  to help grasp the geometric condition ( [ geometric_condition ] ) that is the key for the characterization of the equilibria, we introduce the number that fulfills the remarkable property we first explicit the condition ( [ geometric_condition ] ) on the specific case of the haldane function ( [ haldane ] ) :  is then a root of a polynomial of degree 3 .so there exist at most three solutions of . for small values of , we remark that is very large and has a high slope . on the contrary , for near to , is closed to and has a light slope .intuitively , we expect to have only one root for small values of and three for large values of .for such that there exists a solution of and , one has and , that is is a double root of ( and there exists at most one such double root because is of degree 3 ) . at such ,the hyperbola is tangent to the graph of .intuitively , this corresponds to the limiting case for the parameter in between cases for which there is one or three roots ( see figures [ fig - phi ] and [ fig - phi2 ] where tangent hyperbola are drawn in thick line ) .   when ( on the left ) and ( on the right ) , illustrated with an haldane function ( when ) [ parameters : , , , , , ( left ) / ( right ) ] .[ fig - phi],title="fig:",width=188 ]  when ( on the left ) and ( on the right ) , illustrated with an haldane function ( when ) [ parameters : , , , , , ( left ) / ( right ) ] .[ fig - phi],title="fig:",width=188 ]   illustrated with an haldane function when ( on the left ) and when ( on the right ) [ parameters : , , , ( left ) / ( right ) , ( left ) / ( right ) , ( left ) / ( right ) ] .[ fig - phi2],title="fig:",width=188 ]  illustrated with an haldane function when ( on the left ) and when ( on the right ) [ parameters : , , , ( left ) / ( right ) , ( left ) / ( right ) , ( left ) / ( right ) ] .[ fig - phi2],title="fig:",width=188 ]  to formalize these observations for more general growth functions that fulfill assumptions a1 , we consider the set of at which the hyperbola is tangent to the graph of the function and is locally on one side ( that amounts to have as a local extremum of the function at ) : along with the set   one can distinguish two cases :  1 ._ the single chemostat has only one attracting equilibrium ._ tangencies of the graphs of and could occur for certain values of ( see figure [ fig - phi2 ] as an illustration ) , leading to non - empty set and multi - equilibria .another remarkable feature is that the buffer could create a multiplicity of equilibria .the single chemostat presents a bi - stability ._ the function is necessarily non - monotonic on and a tangency of the graphs of and always occurs for a certain with an abscissa that is located * either at the right of when ( see the right picture of figure [ fig - phi ] ) , * either at the left of when ( see the left picture of figure [ fig - phi ] ) .  in the appendix , more properties on the sets and the multiplicity of equilibria are given in the proposition [ prop1 ] .[ remark1 ] under the conditions of theorem [ main_theo ] , consider the number that guarantees that for any with , the buffered chemostat model admits a unique ( globally asymptotically stable ) positive equilibrium .the map , where is the unique solution of ( [ geometric_condition ] ) on , is clearly continuous and one can then consider the limiting map : when , one has ( resp . ) when ( resp . ) . consider , if it exists , a value of , denoted by , that is such that .although one has for any , there is no reason to have consequently , the map might be discontinuous at such point . in section [ section_example ] ,the non - continuity of the map is illustrated on the haldane function .in this section , we discuss the applications of theorem [ main_theo ] in terms of ecological and biotechnological implications for different buffered configurations .   to better grasp the difference brought by a buffered spatialization compared to a perfectly - mixed environment , we distinguish two main cases depending on the washout if it is an attracting equilibrium or not in the single chemostat .such situation corresponds to cases 1 or 2 of proposition [ prop - chemostat ] :  * either the washout is the only equilibrium ( and is necessarily attracting ) .this happens when the dilution rate is too high or the input concentration too low , that is when one has for any  ] . on the interval , the function is well defined with , and .consequently , there exists at least one solution of , that necessarily belongs to the interval .if , is increasing and there exists a unique solution of whatever is . notice that when the function is increasing on  ] . for any ,there are at least three solutions , that all belong to , by the mean value theorem .+ when , we distinguish two sub - cases :  * : the function is strictly positive on and strictly negative on .furthermore , is decreasing on  ] ( and the set is empty ) . * : the function is strictly positive on  ] . for any , there are at least three solutions , that all belong to the interval , by the mean value theorem .  _ case ii . _ + notice that in this case ( ) the function is non - monotonic .we consider three sub - cases depending on the relative position of with respect to .+ _ sub - case 1 : _ . as for casei , we distinguish :  * : one has and for any . being decreasing on  ] , whatever is .furthermore , this solution has to belong to  ] , and for any  ] for any  ] .+ _ sub - case 2 : _ .one has for any , so there exists a positive equilibrium with . for any  ] , and for , there are at least three solutions by the mean value theorem. +   [ ththieme ] let be an asymptotically autonomous semi - flow with limit semi - flow , and let the orbit have compact closure .then the -limit set is non - empty , compact , connected , invariant and chain - recurrent by the semi - flow and attracts when .   we shall also need to treat a limiting case of the single chemostat that is not covered by proposition [ prop - chemostat ] , when one has exactly for the buffer tank with non - monotonic , that is provided by the following lemma .+  [ lemma ] for any such that and non - negative initial condition with , the solution and of ( [ chemostat2b ] ) is non negative for any and one has   from equations ( [ chemostat2b ] ) one can write the properties and deduces that the variables and remain non negative for any positive time . considering the variable dynamics is , we conclude that are are bounded and satisfy the dynamics of the variable can thus be written as an non autonomous scalar equation : that is asymptotically autonomous .the study of this asymptotic dynamics is straightforward : any trajectory that converges forwardly to the domain  ] ( then the washout is the only equilibrium ) , or when is non - monotonic . in this last situation , none of the cases of proposition [ prop - chemostat ] are fulfilled .we show that for any initial condition such that , the forward trajectory can not converge to the washout equilibrium . from equations ( [ chemostat2b ] )one can write if tends to , then one should have for any finite positive .using taylor - lagrange theorem , there exists a continuous function in such that one can then write note that tends to when tends to .so there exists such that for any , and accordingly to assumptions a1 , there exist positive numbers , such that  ] , that are , ( if ) and ( if ) .the jacobian matrix of the whole dynamics ( [ chemostat2b ] ) at steady state in coordinates is 0 & -\mu(s_{in } ) \end{array } & \begin{array}{cc } f^{\prime}(s_{1}^{\star } ) & \ds d\frac{\alpha(1-r)}{r}\\[3 mm ] 0 & \mu(s_{in})-\alpha d \end{array } \end{array}\right ) \ .\ ] ] when the attraction basin of the washout of the subsystem is not reduced to a singleton , one has necessarily ( see lemma [ lemma ] ) .furthermore , one has and when .so , apart two possible particular values of that are such that or , is non - zero and the equilibrium is thus hyperbolic . finally , we conclude about the possible asymptotic behaviors of the whole dynamics as follows .* the washout equilibrium is attracting when .when , this equilibrium is a saddle ( with a stable manifold of dimension one ) . accordingly to the theorem of the stable manifold , the trajectory solution can not converges to such an equilibrium , excepted from a measure - zero subset of initial conditions .* when , the equilibrium with is always attracting . * when , the equilibrium with is a saddle ( with a stable manifold of dimension one ) . accordingly to the theorem of the stable manifold , the trajectory solution can not converges to such an equilibrium , excepted from a measure - zero subset of initial conditions .when the initial condition of sub - system does not belong to the attraction basin of the washout , proposition [ prop - chemostat ] ensures that converges towards a positive that is equal to or .then , equation ( [ eq ] ) can be equivalently written as : so the dynamics ( [ dynas1 ] ) is asymptotically autonomous with the limiting equation from theorem [ ththieme ] , we conclude that forward trajectories of the sub - system converge asymptotically either to a stationary point where is a zero of the function on the interval , either to the washout point .we show that this last case is not possible . from equations ( [ chemostat2b ] ) , one has and as converges to a positive value , we deduce that can not converges towards . + the functions being analytic for any , the roots are isolated . as for the proof of proposition [ prop1 ]we consider the function that is analytic on its domain of definition and such that this shows that , excepted for some isolated values of in , the zero of are such that .+ let us now write the jacobian matrix of dynamics ( [ chemostat2b ] ) at steady state in coordinates : 0 & -\mu(s_{2}^{\star } ) \end{array } & \begin{array}{cc } f_{r}^{\prime}(s_{1}^{\star})(s_{in}-s_{1}^{\star } ) & \ds d\frac{\alpha(1-r)}{r}\\[3 mm ] 0 & -\mu^{\prime}(s_{2}^{\star})(s_{in}-s_{2}^{\star } ) \end{array } \end{array}\right ) \ .\ ] ] considering the following facts : + i. is hurwitz , + ii . implies that is not equal to .so one has ( cf assumptions a1 ) , + iii . for almost any , + we conclude that any equilibrium is hyperbolic ( for almost any ) and is + - a saddle point when or , + - an exponentially stable critical point otherwise .+ furthermore , the left endpoints of the connected components of the set are exactly the roots of with .finally , from the stable manifold theorem we conclude that , excepted from the stable manifolds of the saddle equilibria , the trajectory converges to an equilibrium that is such that and .this ends the proof of point ii .+  [ prop5 ] assume that the hypotheses a1 are fulfilled with and . there exist buffered configurations with an additional tank of volume that possesses a unique globally exponentially stable positive equilibrium from any initial condition with , exactly when fulfills the condition }\psi(s ) } \ , \ ] ] where the functions and are defined as follows : and is the number the dilution rate has then to satisfy the condition furthermore , one has   * proof of proposition [ prop5 ] .* one can straightforwardly check on equations ( [ chemostat2 ] ) that a positive equilibrium in the first tank has to fulfill let us examine some properties of the function on the interval :  * is negative exactly on the interval , * is negative on with and , * and reaches its maximum on the sub - interval , that is strictly less than ,    the configurations for which there exists a unique solution of the equation ( [ eq - varphi ] ) are exactly those that fulfill the condition , or equivalently with .then , theorem [ main_theo ] with and guarantees that the unique positive equilibrium is globally exponentially stable on the domain .+ among all such configurations , the infimum of can be approached arbitrarily close when is maximizing the function on  ] .

Abstract: we study how a particular spatial structure with a buffer impacts the number of equilibria and their stability in the chemostat model . we show that the occurrence of a buffer can allow a species to persist or on the opposite to go extinct , depending on the characteristics of the buffer . for non - monotonic response functions , we characterize the buffered configurations that make the chemostat dynamics globally asymptotically stable , while this is not possible with single , serial or parallel vessels of the same total volume and input flow . these results are illustrated with the haldane kinetic function . + * key - words . * chemostat , interconnection , multi - stability , global asymptotic stability . + * ams subject classifications . * 92d25 , 34d23 , 93a30 , 90b05 .


in cooperative communication networks , power allocation over channel state may bring rate gains . however , it is not easy to find the optimal power allocation because the exact capacity of most of the wireless networks has not been known .there were some useful cooperation strategies put forward in the literatures which provided efficient approach to transmit information and gave lower bounds for the rate performance of the system .for instance , two relay protocols , decode - and - forward ( df ) and compress - and - forward ( cf ) , were proposed in to evaluate the information rate for relay channels ( rc ) .particularly , the df protocol was shown to be able to achieve the capacity of degraded rc and sender frequency division rc . due to the effectiveness of df and cf , they has been widely used in cooperative communication networks , achieving good rate performance in various networks .based on df and cf protocols , power allocation can be naturally extended to general networks to combat the varying channel states .take the rc as an example again .since there are only three wireless links in the system , it is available for the source and the relay to know the current channel gains before transmissions via timely feedback from the receiver .the global power allocation over fading channel problem in rc has been studied in . by assuming that the source and the relay subject to a sum power constraint, the authors provided algorithms on how to find the optimal power allocation .it is also noted that the power allocation established in achieved the maximal throughput of the relay - receive phase and relay - transmit phase in half - duplex relay channels ( hdrc ) .the result was implicitly based on a buffer at the relay such that if the relay - destination channel is worse , it can store the message and transmit them when the relay - destination channel becomes better . in practice , if the relay has a finite storage and limited processing capability , the system may become unstable and the power allocation gain will degrade .  to improve the achievable rate ,selecting better relay protocol among multiple protocols provides another alternative .this intuition comes from theoretical analysis on combining df and cf in static rc - .it was found superposition structure of the df and cf codewords provides some rate gain with penalties of decoding complexity .moreover , a general insight was also obtained that df outperforms cf for only some of the channel gain combinations while the relationship reverses for the others .this implies that in fading relay channels , a hybrid scheme which selects the better one between df and cf according to the channel state may provide some rate gains while avoiding the complicated codeword design .  instead of using other techniques , e.g. , - , to combat channel fading , in this work ,we thoroughly analyze the relay power allocation over channel state when the relay adopts both df and cf protocols .the remainder of this paper is organized as follows . in sectionii , we introduce the system model and establish a general framework for the relay power allocation problem . in section iii, we present a parameterized form solution for the problem corresponding to df and cf , respectively . in section iv, we further investigate the relay power allocation corresponding to the hybrid scheme and discuss the optimal solution by introducing an auxiliary rate function .let us consider a hdrc as illustrated in fig .[ relay - ch ] . in the figure , , and represent the source , the relay and the destination , respectively .we assume the relay is operated in half - duplex manner . due to the multipath effect , the channel gains are varying along with the time .we assume that the channel gains are holding constant for a fixed time length which is referred to as a block and the channel gains varies independently between consecutive blocks .the signal transmissions in each block are divided into two phases as depicted in fig .[ relay - ch ] . in _ phase 1 _, the source transmits signal while the other two nodes listen . in _ phase 2 _ , the source and the relay transmit signals to the destination . to distinguish the signal in different phases ,let us denote the complex baseband signal transmitted at and received at in _ phase _ by and , respectively . for simplicity , we use and to represent the channel gain variable and its realization for - link in each block .accordingly , transmissions in the hdrc can be expressed as where is additive white gaussian noise ( awgn ) corresponding to in _ phase . for simplicity , we consider that the system is operated in unit bandwidth and we assume that obeys complex gaussian distribution with unit power spectrum density , i.e. , .the source and the relay are assumed to know the channel gains at the beginning of each block . in particular , as the channel phase - shift is well - recovered at the receiver side , we focus on , the distribution of the amplitude of .for the reason of synchronization and power management , we assume that the source transmit signal with the same power and the same time length for the two phases . denote the channel state by . by assuming that the block length is long enough to support one time entire signalling , we can regard the system as a static relay channel for each block .in general , in the static case , the rate performance is a function of the receiver side signal to noise ratio ( snr ) of the three links . to focus on the relay power allocation , we denote the receiver side snr of relay - destination link and the rate function in static hdrc by and , respectively .  in fading hdrc , we consider long time average power constraint at .then the source transmits signal with power regardless of the channel state .however , the relay can adjust adaptively w.r.t the channel state in each block . for clarity, we denote the relay power allocation by .the interest of this paper is to find the optimal power allocation achieving the best rate performance of the system .define . regarding the average rate as the measurement of the rate performance and taking the average power constraint into consideration, we can specify the relay power allocation problem as where ; ; and .  if is concave w.r.t . , one can solve by lagrangian method .consider the lagrangian set .one has it should be noted that only if the rate function is concave w.r.t . should the solution of be the optimal .in this section , we first analyze the concavity of df rate and cf rate . then following necessary condition, we present the optimal power allocation for df and cf based on the inverse function of the derivation of the rates .   the rate achieved by df protocol and cf protocol with gaussian signalingwere presented in proposition 2 and proposition 3 of , respectively . taking a constant fixed source power and the equal - phase assumption into account ,the df rate can be rewritten as where ; represents the shannon formula for complex based model ; represents the correlation coefficient of and .similarly , the cf rate can be expressed as define then we can further express the df rate and cf rate as functions of :   both the df rate and cf rate are concave w.r.t . ._ proof : _first , we analyze the concavity of . in ,the optimal can be found by considering which results in where .note that the first and the second terms in the minimum operation of are monotonically decreasing and increasing w.r.t . ,  ] is a quadratic polynomial w.r.t .then can be expressed as the positive solution of quadratic equation in : ^ 2g_c(s_2)=\frac{t[1+(t+1)s_1]}{2\nu\ln2}.\end{aligned}\ ] ]  according to , is a continuous piecewise function . due tothat is not continuous , analysis on becomes complicated . by comparing the three terms in ,it is not hard to rewrite in a piecewise form as \\ \frac12c(ts_1 ) , & \rho^*<0 \end{cases}\end{aligned}\ ] ]  in fact , if , then is equivalent to that is , if , or equivalently , it arrives that similarly , if , then is equivalent to   after some manipulations , we have   it is easy to verify that =0 ] and =0 ] .accordingly , <r_{df}^{\prime}[f_1 ^ -(s_1)] ] does not exist . according to these analysis, we can define the reverse function of as follows .* if }{2(1+[\sqrt{s_1}+\sqrt{f_1(s_1)}]^2)\ln2},\end{aligned}\ ] ] then is set to the solution of equation in : * if ^ 2)\ln2},\end{aligned}\ ] ] then is set to the solution of equation regarding of  * otherwise , is set to .  with the definition of , one can search for in theorem .this not only helps implementation for power allocation but also provides clues for analyzing the power allocation in combining df and cf protocols .as stated previously , the protocol with selecting a better rate between df and cf can be expressed as then , in a static relay channel , is achievable by switching to the better one between df and cf protocols according to the channel gains .the selection is significant by noting that if , neither df nor cf outperforms the other for all the relay power .define one can easily verify that , if , then and if , then .accordingly , we have   it is noted that =r_{cf}^{\prime}[f(s_1)]\\ > 0=r_{df}^{\prime}[f(s_1)]=r^{\prime}[f^-(s_1)].\end{gathered}\ ] ]  therefore , is not concave w.r.t . anymore . in a fading hdrc, we can not use to find the optimal power allocation corresponding to as what we have done for the case using df / cf protocol . to find some possible solutions ,let us introduce the concave envelops of , .in general , for all and is concave . particularly , for any concave function satisfying for all , one has .  as both and are concave and monotonically increasing functions of , it is easy to deduce that is made up of three parts : a curve coincident with , a line segment connecting two points and another curve coincident with .in particular , the two end points of the line segment should be located on and , respectively . what s more , if is smooth at the end point , the line segment should be tangent with .assume the two end points of the line segment are and , respectively where .then , the slope of the line segment is given by besides , it also has .accordingly , we can express as naturally , the derivation of can be expressed as let us denote the reverse function of by . if , then always holds .therefore , one can define uncountable version of .similar to the definition of , let us define as follows .* if there is a non - empty set satisfying that for each , holds , then is set to the infimum of .* otherwise , set .it can be readily seen from the definition of that the smallest receiver side snr of the relay - destination link , or equivalently , the least relay power , is selected among those satisfying the necessary condition .in fact , for and , this definition of is the same as that of and , respectively .this specific definition of induces a near optimal solution for power allocation based on .we summarize the result in following theorem .[ thm : pa ] given where satisfies .then is a near optimal solution for relay power allocation problem based on which is achieved by selecting the better protocol between df and cf .  _proof : _ similar to what we have done for and , if we use as the static rate performance of the system , we can get an optimal power allocation following from and .   interestingly, the obtained average rate corresponding to also can be achieved by since holds for the solution .this can be verified by noting that holds if and only if .in fact , for any satisfying , it has and .due to the fact that holds in general , the obtained power allocation can guarantee a near - optimal rate performance. investigated relay power allocation over channel state in fading hdrc based on both df protocol and cf protocol . by proving the concavity of the df rate and cf rate ,a parameterized form solution for the optimal power allocation has been presented .furthermore , we considered a hybrid df and cf protocol and introduced an auxiliary function which helped find a near optimal solution of the corresponding relay power allocation problem .d. zhang , p. fan and z. cao . interference cancellation for ofdm systems in presence of overlapped narrow band transmission system " ._ ieee transactions on consumer electronics _ , vol .108 - 114 , 2004 .h. chan , p. fan and z. cao . a utility - based network selection scheme for multiple services in heterogeneous networks " . in _ 2005 proceedings of international conference on wireless networks , communications and mobile computing _ , vol .2 , pp . 1175 - 1180 , 2005 .

Abstract: in fading channels , power allocation over channel state may bring a rate increment compared to the fixed constant power mode . such a rate increment is referred to power allocation gain . it is expected that the power allocation gain varies for different relay protocols . in this paper , decode - and - forward ( df ) and compress - and - forward ( cf ) protocols are considered . we first establish a general framework for relay power allocation of df and cf over channel state in half - duplex relay channels and present the optimal solution for relay power allocation with auxiliary parameters , respectively . then , we reconsider the power allocation problem for one hybrid scheme which always selects the better one between df and cf and obtain a near optimal solution for the hybrid scheme by introducing an auxiliary rate function as well as avoiding the non - concave rate optimization problem .  fading relay channel , power allocation , decode - and - forward ( df ) , compress - and - forward ( cf ) .


in recent years , smart grid has attracted significant interest in both communities of communications and power systems . in a smart grid ,modern communication technologies are used to convey information like system parameters ( voltage , frequency , harmonics , etc ) and power consumption information in order to improve the robustness , agility and efficiency of power grid .for example , as illustrated in fig .[ fig : illu ] , phasor measurement units ( pmus ) send report to the power plant which takes actions to stabilize the power grid .however , the communication infrastructure also brings vulnerability to the smart grid .an attacker can attack the communication links using various approaches such as jamming in the physical layer and byzantine attack in the upper layers .the attack could result in delay or drop of report packets .the attacker may revise the report such that the received report is wrong , thus possibly incurring system instability and even large area blackout which brings the loss of millions of dollars .therefore , a secure design of smart grid is in a pressing need .  in this paper, we study the trustiness framework based secure control in smart grid .we assume that each report could be substituted by a false report from an attacker . instead of studying the security protocols in the communication networks , which has been intensively studied for data networks like internet , we focus on the controller side , i.e. , the power plant , which is aware of possible attacks .the controller can predict the future system state and then evaluate the trustiness of reports from different pmus .based on these trustiness , the controller takes a corresponding control strategy , e.g. , dropping untrusted packets .the trustiness will also be fed back to the pmus such that the pmus can adjust their security setups ( e.g. , the keys or the cryptography protocols ) .the system is illustrated in fig .[ fig : illu ] . for example, the reports from pmu a is attacked by the attacker .then , the trustiness of the reports from pmu a is reduced , which may make the controller drop the corresponding reports .meanwhile , pmu a may change its key or use a more secure protocol when it finds out that its trustiness has been significantly decreased .note that the reliability issues for control systems have been considered in supervisory control and data acquisition ( scada ) standard . however , scada is mostly focused on the reliability subject to random failures , instead of malicious attacks .there are some studies on the control systems subject to malicious attacks . and introduce general problems and approaches for secure control without exploring the details . is focused on the control system subject to denial - of - service ( dos ) attacks , thus mainly addressing the packet delays or losses .there have not been any studies on combating the proofing attacks , particularly , applying the trustiness framework in the secure control .note that the trustiness system has been widely used in information systems , e.g. , .however , they are not designed for control systems .the unique system dynamics of the control system can be exploited to build the corresponding effective trustiness . in this paper , we apply kalman filtering to predict the system state using different combinations of pmu reports , thus realizing a cross - check of the report trustiness .  the remainder of this paper is organized as follows . the system model of the controller and communication infrastructure is introduced in section [ sec : system ] .a mechanism of evaluating the trustiness of different pmus is proposed in section [ sec : evaluation ] .the numerical simulation results and conclusions are provided in sections [ sec : numerical ] and [ sec : conclusion ] .in this section , we first introduce the linear system model for power grid . then , we explain the lqr criterion of the control .   we model the dynamics of power grid as a discrete - time linear system , whosedynamics are given by where is an -vector and represents the system state of the power grid , is an -vector representing the observations and is the action taken by the controller .the matrices , and are specified by the system .both and are gaussian noise . for simplicity, we suppose that each dimension of the observation vector is sensed by a pmu .it is easy to extend to the general case in which the sensor reports have overlaps .each pmu sends its observations to the controller via a communication channel since they are not located at the same place as the controller .we use the following assumptions throughout the paper :  * each report can be successfully received by the controller if there is no attack .this is reasonable for communication channels with good qualities .for the practical case of occasional packet drop , the control strategy can be obtained by considering the corresponding element in the observation matrix as zero . * each report could be replaced with a false report by an intervening attacker .the attacker could intercept the report and insert its own one .however , we assume that not all reports are replaced . * for simplicity, we assume that there is at most one attacker .the principle of the trustiness system can be extended to the case of multiple attackers at the cost of more computational cost .   when there is no attacker , we assume that the controller adopts the lqr control with an infinite time horizon with the cost function given by ,\end{aligned}\ ] ] where and are positive definite matrices .the physical meaning of the objective function is given below :  * the term is the norm ( with respect to the positive definite matrix ) of the system state vector . *the term is the norm of the action vector with respect to the matrix , which represents the cost due to the action itself .based on the cost function in ( [ eq : lqg ] ) , the lqr action is given by where is the estimation of the system state fed back from a state estimator which will be explained later , and and the matrix satisfies the algebraic riccati equation , which is given by this section , we propose a mechanism for evaluating the trustiness of each sensor .the essential reason that the controller can evaluate the trustiness of each sensor is that the controller can predict the future state with some uncertainty .if the report from a sensor is significantly deviated from the prediction , then the controller can consider this sensor as unreliable and ignores its reports .the basic principle is to evaluate the trustiness of each pmu by comparing its report with the prediction obtained from the reports of other pmus .the procedure is illustrated in fig .[ fig : kalman ] .     in each time slot, the controller can predict the system state in the next time slot , i.e. , , according to its own action and the current system state . since it is possiblethat there is one attacker , the controller computes predictions by excluding one sensor in each prediction .the kalman filtering can be applied for the system state estimation . according to , the system state is gaussian distributed .when the observation from pmu is excluded , the expectation of the system state is given by where where is obtained by removing the -th row from the observation matrix , and and covariance matrix given by where    using the prediction obtained from the reports of pmus except pmu , the _ a posteriori _ probability of , i.e. , ( here means the observations excluding that of pmu ) , is gaussian distributed with the expectation given by where is the -th row in matrix , and variance   we denote by the type of pmu . if pmu is an attacker ; otherwise .then , we define the suspicious level of pmu as the following conditional probability , which is given by the trustiness can be defined as .the challenges for computing the suspicious level is the unknown attacking strategy .we first assume that there must be an attacker . using the bayesian rule, we have where the last approximation is obtained by decomposing the joint distribution into the product of the probabilities in each time slot .note that this approximation is not rigorous .however , it simplifies the computation and the validity will be demonstrated in the numerical simulations .we do the following further simplification : by assuming the independence among the pmus .although this assumption does not hold , it simplifies the analysis .we further assume that is a constant since we have no knowledge about the attacker s strategy . substituting the approximation in ( [ eq : approx ] ) into ( [ eq : trust ] ) , we obtain we then approximate the probability by .when it is possible that there is no attacker , it is easy to repeat the above procedure and obtain the following approximation for the suspicious level of pmu , which is given by where represents the _ a priori _ likelihood that there is no attacker , which is given by obviously , the large is , the less sensitive the controller is to possible attackers .one approach to handle the attacker is to omit its reports once determining that it is an attacker .alternatively , we propose a heuristic approach for the control based on the trustiness values , called _weighted prediction_. suppose that we still use the lqr control .then , the control action taken at time is given by where the system state estimation is a function dependent on the suspicious levels of different pmus .we set where the estimation is the weighted sum of the system state estimations of different excluded pmus .when the suspicious level of pmu is high , the corresponding system state estimation , which excludes the reports from pmu , will dominate ( recall that is obtained by excluding the reports from pmu ) .the proposed algorithms are summarized in procedure [ alg : trustiness ] .in this section , we use numerical simulations to demonstrate the proposed trustiness system in smart grid .we adopt the linear model analyzed in example 6.2 of , in which the system is described using the following continuous - time linear dynamics : where the matrix is given by    and the matrix is given in ( [ eq : mat_k ] ) ( at the top of the next page ) .the details of the model can be found in .  since we discuss the discrete - time model in this paper , we approximate the continuous - time model by setting a small time step , which is given by therefore , we assume the following discrete - time model : where we ignore the step in the index .we assume that the pmus can observe the system state directly , i.e. , , each pmu for one dimension .we further assume that pmu 1 is malicious while all other pmus are honest .   in figures [ fig : attack ] and [ fig : attack2 ] , the evolution of the suspicious level is shown for the attacker and two honest pmus . in fig .[ fig : attack ] , we assume that the report of the attacker is a gaussian random variable with zero mean and variance 0.1 . the attacker decides to attack or not to attack with probability 0.2 ( called _ attack frequency _ ) .we observe that there is some fluctuation at the beginning .then , at the times when the attacker launches attacks , the suspicious level of the attacker increases significantly .after around 80 time slots , the attacker can be well distinguished from the two honest pmus . in fig .[ fig : attack2 ] , we assume that the attacker attaches a gaussian noise with zero expectation and variance 0.1 to the observation . the attack frequency is increased to 0.5 .we observe that the suspicious level increases more smoothly .       in fig .[ fig : cdf ] , we plot the cumulative distribution function ( cdf ) curves of the time when the controller claims that an attacker is detected .we assume that the controller claims the attacker when the suspicious level is larger than 0.7 .the cases of false alarms are excluded ( note that a false alarm is defined as the event that a honest pmu is claimed to be an attacker ) .we change the attack frequencies from 0.1 to 0.3 .the linear system runs for 200 time slots .we observe that , when the attack frequency is 0.1 , the attacker is not detected within the 200 time slots in around 40% cases .when the attack frequency is increased , the attacker will be detected faster .    in fig .[ fig : roc ] , we plot the receiver operation characteristic ( roc ) curves , which show the average delay of detection ( excluding the false alarms ) and false alarm rate .the attack frequencies are 0.2 and 0.4 , respectively .again , we use the threshold of 0.7 for the suspicious level .obviously , the detection delay increases when the attack frequency becomes larger with a fixed false alarm rate .     in figures [ fig : cost1 ] and [ fig :cost2 ] , the cost averaged over 2000 time slots and 100 realizations is shown .we assume that and in ( [ eq : lqg ] ) , i.e. , we pay much more attention to the norm of the system state .we compare the costs using the weighted system state estimation in section [ sec : evaluation ] and the system without any counter measure for the attacker ( i.e. , full trust to each pmu ) .  in our test, we found that a small amplitude attack ( e.g. , the attacker uses the same attack as in fig .[ fig : cdf ] ) causes very small impact on the system .this is because that the kalman filter has certain inherent robustness since the attack can be partially mitigated by observations from other pmus .therefore , we assume that the report of pmu1 is the sum of the original report and a strong noise with a large variance , which is called _ attack amplitude_. in fig .[ fig : cost1 ] , we assume that the attack amplitude is 100 and change the attack probabilities .we observe that , as the attack probabilities increases , the average cost of the full trust case is significantly increased .meanwhile , the total cost is decreased as the attack frequency increases .a possible reason is that a higher attack frequency may cause a more rapid degradation of the trustiness of the attacker .the total cost is also shown for different attack amplitudes in fig .[ fig : cost2 ] .again , the total cost increases as the attack amplitude is increased in the full trust case . in a contrast to fig . [fig : cost1 ] , the average cost of the weighted system state case is not a monotonic function of the attack amplitude .the reason could be : when the attack amplitude is small , the attacker causes little damage to the system ; when the amplitude is large , the controller can detect the attacker early and avoid the cost in later time slots .in this paper , we have analyzed the possible spoof attack on the smart grid system , in which the attacker can intercept and true report and send its faked report to the controller , which can cause severe damage to the power grid .we have proposed a trustiness system for the controller , in which multiple kalman filtering processes , with each pmu excluded , are used to cross check the suspiciousness of each pmu .the suspicious levels are then used as weights for the system state estimation for the lqr control .numerical simulations have shown that the attacker can be effectively detected and the weighted system prediction approach significantly outperforms the system unaware of possible attacks .               c. w. taylor , d. c. erickson , k. e. martin , r. e. wilson and v. venkatasubramanian , `` wacs - wide - are stability and voltage control system : r&d and online demonstration , '' _ proceedings of the ieee _ , pp.892906 , may 2005 .

Abstract: smart grid , equipped with modern communication infrastructures , is subject to possible cyber attacks . particularly , false report attacks which replace the sensor reports with fraud ones may cause the instability of the whole power grid or even result in a large area blackout . in this paper , a trustiness system is introduced to the controller , who computes the trustiness of different sensors by comparing its prediction , obtained from kalman filtering , on the system state with the reports from sensor . the trustiness mechanism is discussed and analyzed for the linear quadratic regulation ( lqr ) controller . numerical simulations show that the trustiness system can effectively combat the cyber attacks to smart grid .


the energy concentration problem in the time - frequency domain plays a crucial role in signal processing .the foundation of this problem comes from 1960s the research group of bell labs .the problem states that for any given signal with its fourier transform ( ft ) the energy ratios of the duration and bandwidth limiting of the signal , i.e. , and of both in fixed time  ] domains , satisfy the following inequality let be the total energy of . by the parseval theorem , the energy in time and frequency domains are equal , i.e. , . without loss of generality, we consider the unit energy signals throughout this paper , i.e. , .  the important constant in eq . ( [ eq.bound ] ) is the eigenvalue of the zero order prolate spheroidal wave functions ( pswfs ) .the pswfs are originally used to solve the helmhotz equation in prolate spheroidal coordinates by means of separation of variables . in 1960s , slepian _ et al ._ found that pswfs are solutions for the energy concentration problem of bandlimited signals .their real - valued pswfs are solutions of the integral equation where are eigenvalues of pswfs . here ] are the fixed time and frequency domains , respectively .important properties of pswfs are given in .the following properties follow form the general theory of integral equations and are stated without proof .1 .  eq.([eq.1dpswfs ] ) has solutions only for real , positive values eigenvalues .these values is a monotonically decreasing sequence , , such that .2 .  to each there corresponds only one eigenfunction with a constant factor .the functions form a real orthonormal set in ;\r) ] and the spatial domain \times[-\tau,\tau] ] , if qlcts are  degenerated to 2d fourier transform ( ft ) , then qpswfs becomes the 2d  real pswfs , which is given by   here , if is separable , i.e. ,   , then the 2d pswfs  can be regarded as the product of two 1d pswfs . to aid the reader ,see   for more complete accounts of this subject .( ii ) : :  in a unit disk , the qlcts are degenerated to 2d ft , then the qpswfs  between the circular pswfs   we call the right - hand side of eq .( [ eq.qpswf ] ) is the finite qlcts .however , only for the .there is a scale factor added to the parameter matrix , which is different from the definition of qlcts.   some important properties of qpswfs will be considered in this part , which are crucial in solving the energy concentration problem .let be the qpswfs associated with their qlcts and .then are solutions of the following integral equation where for are the eigenvalues corresponding to and , , for , and .( [ eq.lowpass ] ) is named the _ low - pass filtering form of qpswfs associated with qlcts_.  we shall show that eq .( [ eq.lowpass ] ) is derived by the eq .( [ eq.qpswf ] ) .straightforward computations of the right - hand side of eq .( [ eq.lowpass ] ) show that applying the following two important equations to the last integral , then we have ^{\mathbf{j}v_2v}dv_1dv_2.\end{aligned}\ ] ] combining eq.([eq.qpswf ] ) with the parameter matrices , and , we have ^{\mathbf{j}v_2v}dv_1dv_2\\ & = & \frac{1}{2\pi}\int_{\bm{\sigma}}e^{\mathbf{i}v_1u } \left [ \lambda_n e^{\mathbf{-i}\frac{cd_1}{2b_1}(\frac{b_1v_1}{c})^2 } \mathbf{i}^{n } \sqrt{cb_1 } \bm{\psi}_{n}\big(\frac{b_1v_1}{cb_1},\frac{b_2v_2}{cb_2}\big ) \sqrt{cb_2 } \mathbf{j}^{n } e^{\mathbf{-j}\frac{cd_2}{2b_2}(\frac{b_2v_2}{c})^2 } \right ] e^{\mathbf{j}v_2v}dv_1dv_2\\ & = & \frac{1}{2\pi } \lambda_n \int_{\bm{\sigma}}e^{\mathbf{i}v_1u } \left[e^{\mathbf{-i}\frac{cd_1b_1}{2}(\frac{v_1}{c})^2 } \mathbf{i}^{n } \sqrt{cb_1 } \bm{\psi}_{n}\big(\frac{v_1}{c},\frac{v_2}{c}\big ) \sqrt{cb_2 } \mathbf{j}^{n } e^{\mathbf{-j } \frac{cd_2b_2}{2}(\frac{v_2}{c})^2 } \right ] e^{\mathbf{j}v_2v}dv_1dv_2\\ & = & \frac{1}{2\pi } \lambda_n c^3\sqrt{b_1b_2}\mathbf{i}^{n } \left[\int_{\bm{\tau } } e^{\mathbf{i}w_1cu } e^{\mathbf{-i}\frac{cd_1b_1}{2}w_1 ^ 2 } \bm{\psi}_{n}(w_1,w_2 ) e^{\mathbf{-j}\frac{cd_2b_2}{2}w_2 ^ 2 } e^{\mathbf{j}w_2cv}dw_1dw_2 \right ] \mathbf{j}^{n}\\ & = & \lambda_n c^3\sqrt{b_1b_2}\mathbf{i}^{n } \left [ \lambda_n e^{-\mathbf{i}\frac{c(\frac{-a_1}{{b_1}^2})}{2b_1}(ub_1)^2 } \frac{(-\mathbf{i})^n}{\sqrt{\mathbf{i } } } \sqrt{cb_1\mathbf{i } } \bm{\psi}_{n}\big(\frac{ub_1}{b_1},\frac{vb_2}{b_2}\big ) \sqrt{cb_2\mathbf{j } } \frac{(-\mathbf{j})^n}{\sqrt{\mathbf{j } } } e^{-\mathbf{j}\frac{c(\frac{-a_2}{{b_2}^2})}{2b_2}(vb_2)^2 } \right ] \mathbf{j}^{n}\\ & = & \lambda_n^2 c^4b_1b_2e^{\mathbf{i}\frac{ca_1}{2b_1}u^2 } \bm{\psi}_{n}(u , v)e^{\mathbf{j}\frac{ca_2}{2b_2}v^2 } = c^4b_1b_2\lambda_n^2 \tilde{\bm{\psi}}_{n}(u , v)=:\mu_n \tilde{\bm{\psi}}_{n}(u , v).\end{aligned}\ ] ] the proof is complete .  for the specific parameters , eq .( [ eq.lowpass ] ) becomes the low - pass form of qpswfs associated with qft   to obtain the following property , we shall show a special convolution theorem of any -valued signal and real - valued signal .[ lem.convolution ] let and associated with their qft and with , where , and .the convolution of and is defined as then the qft for holds   let and , straightforward computation the qft in eq.([eq.2sideqft ] ) of the convolution between and shows that g(m , n ) e^{-\mathbf{j}yv}e^{-\mathbf{j}nv}dxdydmdn.\end{aligned}\ ] ] with , the last integral becomes since we have known is real - valued , then we have this completes the proof .note that if the real signal with is real valued , then .it means that   the convolution theorems for quaternion fourier transform was given in .lemma [ lem.convolution ] is following the idea of theorem 13 and lemma 14 in . for completeness, we proof the convolution formula in eq .( [ eq.convolution ] ) .let be the qpswfs associated with qlcts and , satisfies which extends the integral of from to .let eq .( [ eq.lowpass ] ) is actually a convolution of with two - dimensional _ sinc _ kernel as follows denote . from lemma [ lem.convolution ] , let , the and its qft is real valued function .then taking qft to the both sides of eq .( [ eq.convl1 ] ) , we have immediately , we obtain that for , i.e. , here from lemma [ lem.convolution ] , taking the inverse qft on both sides of the above equation , it follows that satisfies eq .( [ eq.allpass ] ) , which extends the integral domain of from to .the propositions [ pro.eigenvalues ] and [ pro.orthogonalintau ] follow from the general theory of integral equations of hermitian kernel and are stated without proof .[ pro.eigenvalues ] eq .( [ eq.lowpass ] ) has solutions for real or complex .these values are a monotonically decreasing sequence , , and satisfy .  [ pro.orthogonalintau ] for different eigenvalues , the corresponding eigenfunctions are an orthonormal set in , i.e. ,   the eigenfunctions form an orthonormal system in , i.e. ,   combining eq .( [ eq.lowpass ] ) , the orthogonality in can be immediately deduced as follows the present section , we will consider the energy concentration problem of bandlimited -valued signals in fixed spatial and qlcts - frequency domains .the definitions and notations of bandlimited -valued signals associated with qlcts and qft are introduced in the following .an -valued signal with finite energy is -bandlimited associated with qlcts , if its qlcts vanishes for all outside the region , i.e. , denote the set of -bandlimited -valued signals associated with qlcts , i.e. ,   an -valued signal with finite energy is -bandlimited associated with qft , if its qft vanishes for all outside the region , i.e. , denote the set of the -bandlimited -valued signals associated with qft , i.e. ,   note that the relationship between qlct and qft for an -valued signal  that is to say if , then for the , is also in , that means \times [ \frac{-\sigma}{b_2},\frac{\sigma}{b_2}]= :  \tilde{\bm{\sigma}}.  \end{aligned}\ ] ] then , because   now we pay attention to the energy concentration problem associated with qlcts . to be specific, the energy concentration problem associated with qlcts aims to obtain the relationship of the following two energy ratios for any -valued signal with finite energy in a fixed spatial and qlcts - frequency domains , i.e. , and , by the parseval identity in eq .( [ eq.parsevalqft ] ) , the two ratios can also be obtained by note that the value of and are real values in $ ] .   in this part, we only consider the energy problem for , i.e. , and .concretely speaking , given an unit energy , the energy concentration problem is finding the maximum of , i.e. , denote the maximum as follows   let , we can also reformulate as follows we conclude that the maximum can be taken if . to derive this fact , the generally cross - correlation function of and was introduced at first , consider the , we have from the complex - valued schwarz s inequality , the takes the maximum value if , where is a constant .similarly , we can define the cross - correlation function of -valued signals and as follows since the quaternionic schwarz s inequality also holds .then to get the maximum value of , the relationship between and satisfies , where is a constant . here, we find that to achieve the maximum , the two functions should be the same except a constant factor .for this reason , there exists a constant such that .let and are the qft for and , respectively .taking qft to both sides of the equation , we have since , then is also in , i.e. , from lemma [ lem.convolution ] , taking the inverse qft to the above equation , we have substituting to the above equation , we have which is the low - pass filter form of qpswfs .now we show that -bandlimited -valued signals satisfying the low - pass filter form eq .( [ eq.low ] ) can reach the maximum .[ th.theoremexist ] if the eigenvalues of the integral equation have a maximum , then .the eigenfunction corresponding to is the function such that are reached .  for any -bandlimited signal ,construct a function as follows let the qft of as , it follows that it means that .denote the energy ratio for in the fixed spatial domain as follows we conclude that for any , can not exceed the .direct computations show that the energy of the signal is given as follows \\ & = & { \bf sc } \left [ \int_{\r^2}\bm{\tilde{f}}_{\bm{\tau}}(x , y)\bm{\tilde{s}}(x , y)dxdy \right]\\ & = & { \bf sc } \left [ \int_{\r^2}p_{\bm{\tau}}(x , y)\bm{\tilde{f}}(x , y)\overline{\bm{\tilde{s}}(x , y)}dxdy\right].\end{aligned}\ ] ]  on the other hand , we consider that \\ & = & { \bf sc } \left [ \int_{\tilde{\bm{\sigma}}}\f(\bm{\tilde{s}})(u , v ) \overline{\f(\bm{\tilde{f}})(u , v)}dudv \right].\end{aligned}\ ] ] since and \right|^2 \leq \left|\int_{\tilde{\bm{\sigma}}}\f(\bm{\tilde{s}})(u , v ) \overline{\f(\bm{\tilde{f}})(u , v)}dudv\right|^2,\end{aligned}\ ] ] simplifying the above three inequalities , we obtain that from which it follows that we also have the following result for \right|^2\\  \nonumber & \leq & \left|\int_{\r^2}p_{\bm{\tau}}(x , y)\bm{\tilde{f}}(x , y)\overline{\bm{\tilde{s}}(x , y)}dxdy\right|^2\\  \nonumber & \leq & \int_{\r^2}p_{\bm{\tau}}(x , y)\bm{\tilde{f}}(x , y)\overline{\bm{\tilde{f}}(x , y)}dxdy \int_{\r^2}p_{\bm{\tau}}(x , y)\bm{\tilde{s}}(x , y)\overline{\bm{\tilde{s}}(x , y)}dxdy.\end{aligned}\ ] ] here , we take into two parts , i.e. , , and use the schwarz inequality for the above inequality .clearly , , then summarizing , we have that means , for any , .  if , then eq . ( [ eq.1 ] ) and eq .( [ eq.2 ] ) must be equalities .this is attained only by setting with .it means that is an eigenfunction of eq .( [ eq.exist ] ) and is the corresponding eigenvalue , i.e. , .  at last , we will show that , and the eigenfunction corresponding to is the function such that is reached . by definition of , there exists a maximum and we denote the maximum as and the corresponding signal as .as we have shown , the corresponding the eigenfunction satisfies .here , corresponds to the maximum eigenvalue of .hence , .  in order to prove that , it suffices to show that is an eigenfunction of the integral equation eq .( [ eq.exist ] ) , or equivalently , that with defined as in eq .( [ eq.s ] ) with .obviously , and , because is maximum by assumption .the proof is complete .theorem [ th.theoremexist ] shows that for arbitrary unit energy -bandlimited -valued signal associated with qlcts the maximum value of can be achieved by the qpswfs .in fact , from the symmetry theorem of fourier theory , there is also a similar integral equation for time - limited signals , which have the maximum .the prove of this conclusion is similar to theorem [ th.theoremexist ] .if the eigenvalues of the integral equation have a maximum , then have a maximum number and .the eigenfunction corresponding to is the function such that are reached .( [ eq.fptauf ] ) is equivalent to eq .( [ eq.exist ] ) with and .   in this section, we will discuss the relationship of in eq .( [ eq.extremal ] ) from three cases :  * is a -bandlimited signal associated with qlcts .* is a -time - limited signal .* is an arbitrary signal .the first case follows form the general theory of the in section [ s5.1 ] .as we have known is in when , i.e. , . from theorem [ th.theoremexist] , we know that the maximum equals the maximum eigenvalue in eq .( [ eq.exist ] ) . using the expansion for the , where .it is clear that hence , . if , then . if , then we can find a signal whose energy ratio in spatial domain equals , and in this case , is not unique .the second case means . from the property of symmetry of the qlctwe conclude that all the properties for signals have corresponding time - limited counterparts . reversing and ,we conclude that . specially , if , then .  for the third case ,considering arbitrary signals with , we aim to find the maximum and the corresponding signal .if , as we noted in the case of , we can find with energy ration , hence , .therefore , we only need to consider the case of .[ th.maxbeta ] the maximum of must satisfy the following equation where is the largest eigenvalues of eq .( [ eq.lowpass ] ) and the corresponding for the maximum is given by   before giving the proof to eq .( [ eq.maxbeta ] ) , we first need to present the following fact . given a function with spatial projection and frequency projection , we construct a new function as follows where and are two constants such that the energy of is minimum , where denote , and , the energy ratios for and as eq .( [ eq.extremal ] ) , respectively .we conclude that , .suppose the energy of equals to and we rewrite as follows from the orthogonality principle , it follows that which means .meanwhile , we have of by now we denote two energy for the projection of as follows the and will be simply written as and in the following , respectively . since , we have therefore , and .that means , in order to get the maximum , we can formula a function as follows taking qft to both sides for eq .( [ eq.combi ] ) and then taking frequency projection , we have rearranging this formula , we obtain that .\end{aligned}\ ] ] taking inverse qft to the above equation , we have on the other hand , taking the spatial projection to eq .( [ eq.combi ] ) , we get rearranging this equation , it becomes taking qft on both sides to the above equation , it follows that applying eq .( [ eq.inversepafts5 ] ) and eq .( [ eq.pafts5 ] ) , we have simplifying the above equality , we obtain that from above equality , we find that is one of qpswfs for eq .( [ eq.lowpass ] ) and the corresponding eigenvalue is . by the relationship between and in eq .( [ eq.fandf5 ] ) , we conclude that in eq .( [ eq.combi ] ) can be rewritten as now , we compute the inner product of the above equation with and respectively . since for , we have then we have and .it follows that with and , the parameters become and .that means from which it follows that in order to get the maximal , we must take the largest .the corresponding function is the proof is complete .  until now we have discussed all the relationships of , as well as the signals to reach the maximum value of for different conditions of .-bandlimited associated with qft and the modulus of in time and qft - frequency domains ., width=566 ]  -bandlimited associated with qlcts and the modulus of in time and qlct - frequency domains with ., width=566 ]  now we give some comparison examples to intuitively illustrate the concentration levels of qpswfs associated with qlcts .the widely used gaussian function will be compared with qpswfs . in theorem[ th.theoremexist ] , we have shown that qpswfs are the most energy concentred -bandlimited signals .  now , a -bandlimited gaussian function is constructed at first .consider the truncated gaussian function in qlcts - frequency domain as follows where is the qlct of .obviously , has unit energy .this -bandlimited gaussian function in spatial domain becomes as for the qpswfs , by means of the classical one - dimensional pswfs of zero order we now construct a special qpswf as follows where is the first one - dimensional zero order pswf . here , we construct the qpswf under the condition of .the qlcts for the qpswf becomes for both of the -bandlimited signals above , the energy ratios equal to in qlct - frequency domain .the energy ratio pair in spatial and frequency in the comparison is noted as .-time - limited associated with qft and the modulus of in time and qft - frequency domains . , width=566 ]  -time - limited associated with qlct and the modulus of in time and qlct - frequency domains with . , width=566 ]  in fig .[ fig.beta1 ] and fig .[ fig.beta2 ] , we will show two pairs of the energy ratios for and in spatial domain associated with qlct with two kinds of different parameter matrices . in fig .[ fig.beta1 ] we set the parameter matrices of qlct , , which is already a qft . in this case , the energy ratios for and are very close .however , in fig .[ fig.beta2 ] we set the parameter matrices of qlct , . in this case , the energy ratio for is and for is . in fact , we just change the parameters , from to .that means , for qpswfs the energy is more concentred then truncated gaussian function .  as for the time - limited function , there are the similar results like -bandlimited cases .we also list two pairs of the energy ratios for and in qlct - frequency domains in fig .[ fig.xi1 ] and fig .[ fig.xi2 ] . in fig .[ fig.xi1 ] we also set the parameter matrices of qlct to be the qft .the parameter matrices of qlct in fig .[ fig.xi2 ] is the same as that in fig .[ fig.beta2 ] . in this two pair cases, you may see the energy ratios for and are very close .but one more thing different from fig .[ fig.beta1 ] and fig .[ fig.beta2 ] is that the energy ratios for and associated with qft is smaller than the energy ratios for and associated with the second parameter matrices .that means , the parameter matrices of qlct is vary important . in some sense , for specific conditions the results for qlct will be better than qft .this paper presented a new generalization of pswfs , namely qpswfs , which are the optimal -valued signals for the energy concentration problem associated with the qlcts .we developed the definition of the qpswfs associated with qlcts and established various properties of them . in order to find the energy distribution of for any -valued signals , we not only derive the parseval identity associated with ( two - sided ) qlcts , butalso show that the maximum for -bandlimited signals associated with qlcts in a fixed spatial domain must be qpswfs .the authors acknowledges financial support from the national natural science foundation of china under grant ( no .11401606,11501015 ) , university of macau ( no . myrg2015 - 00058-fst and no . myrg099(y1-l2)-fst13-kki ) and the macao science and technology development fund ( no . fdct/094/2011/a and no .fdct/099/2012/a3 ) .10 url # 1`#1`urlprefixhref # 1#2#2 # 1#1        h. j. landau and h.o . pollak ._ prolate spheroidal wave functions , fourier analysis and uncertainty iii : the dimension of space of essentially time - and bandlimited signals _ , bell system technical journal , 41(4 ) , 12951336 ( 1962 ) .d. slepian ._ prolate spheroidal wave functions , fourier analysis and uncertainty iv : extensions to many dimensions ; generalized prolate spheroidal functions _ , bell system technical journal , 43(6 ) , 30093057 ( 1964 ) .j. j. ding and s. c. pei ._ reducing sampling error by prolate spheroidal wave functions and fractional fourier transform _ , in proceedings of the ieee international conference on acoustics , speech , and signal processing , 217220 ( 2005 ) .j. morais , k. kou , and y. zhang ._ generalized prolate spheroidal wave functions for offset linear canonical transform in clifford analysis _ , mathematical methods in the applied sciences , 36(9 ) , 10281041 ( 2013 ) .e. b. corrochano , n. trujillo , and m. naranjo ._ quaternion fourier descriptors for preprocessing and recognition of spoken words using images of spatiotemporal representations _ , mathematical imaging and vision , 28 , 179190 ( 2007 ) .p. bas , n. lebihan , and j. m. chassery ._ color image water marking using quaternion fourier transform _ , in proceedings of the ieee international conference on acoustics , speech and signal processing , 521524 ( 2003 ) .

Abstract: quaternionic linear canonical transforms ( qlcts ) are a family of integral transforms , which generalized the quaternionic fourier transform and quaternionic fractional fourier transform . in this paper , we extend the energy concentration problem for 2d hypercomplex signals ( especially quaternionic signals ) . the most energy concentrated signals both in 2d spatial and quaternionic linear canonical frequency domains simultaneously are recently recognized to be the quaternionic prolate spheroidal wave functions ( qpswfs ) . the improved definitions of qpswfs are studied which gave reasonable properties . the purpose of this paper is to understand the measurements of energy concentration in the 2d spatial and quaternionic linear canonical frequency domains . examples of energy concentrated ratios between the truncated gaussian function and qpswfs intuitively illustrate that qpswfs are more energy concentrated signals .   quaternionic linear canonical transforms , energy concentration , quaternionic fourier transform , quaternionic prolate spheroidal wave functions .