landauer s principle [ [ landuer1 ] ] states that any logically irreversible operation on information , such as the erasure of a bit or the merging of two computation paths , is always associated with an increase of entropy of the non - information bearing degrees of freedom of the information processing apparatus or its environment [ [ bennet ] ] and consequently each bit of lost information will lead to the release of at least amount of heat .but it is well known that most of the commonly used classical gates ( except not and identity ) are irreversible and they erase at least one bit of information in every operation .thus the irreversible logic gates will always release some heat energy . on the other hand ,moore s law states that the number of transistors in a chip gets doubled in every 18 months .therefore , if we continue to design chips with the help of conventional irreversible logic gates then the lower limit of power loss will continue to increase .this led to the idea of reversible computation and reversible logic .since quantum mechanics is essentially reversible , quantum mechanical processes appeared as good candidate to construct reversible gates and these gates are known as quantum gates . when we glue some of these gates we obtain a quantum circuit .after introduction of the idea of quantum computation it has already been seen that there exist some quantum algorithms [ [ the : nielsen and chuang ] ] which works much faster than their classical counter parts , there exists infinitely secured quantum cryptographic protocol [ [ crypto1 ] , [ crypto2 ] ] ; there exists protocol for quantum teleportation [ [ the : cerf - teleportation ] ] and all these processes which establish quantum computing as a superior future technology involves quantum circuits and quantum gates .it has also been seen that it is possible to design classical reversible gates and classical reversible circuits but since they can not handle superposition of states ( qubit ) they just form special cases of quantum circuit or a subset of the set of the quantum circuits . but from the construction point of view they are easy to build . keeping this background in mind it is reasonable to state that all novel applications of reversible / quantum computation essentially involve reversible / quantum circuits .a considerable amount of work has already been done in the field of designing and optimization of reversible combinatorial circuit [ [ maslov1]-[ad vos ] ] .but the designing aspect of reversible sequential circuit is not yet studied rigorously .this is because of the fact that feedback in a reversible circuit can not be visualized in the usual sense in which feedback is visualized in a conventional irreversible circuit .this issue was first addressed by toffoli [ [ toffoli ] ] . in [ [ toffoli ] ] he had shown that the reversible sequential circuits can be constructed provided the transition function of the circuit block without the feedback loop is unitary . his ideas on the sequential reversible circuit had further strengthen in his pioneering work on conservative logic [ [ the : toffoli - conservative logic ] ] . later on some effortshave been made to construct reversible sequential circuit [ [ picton]-[thapiyal2 ] ] .all these efforts are concentrated on the designing of various flip flops because of the fact that the flip flops are the basic building block of the memory element of a computer and if one wishes to build a reversible classical computer then these designs will play a crucial role . butseveral conceptual issues related to designing and optimization of sequential circuits are not addressed till now . in the next sectionwe address those conceptual issues related to the feedback and the choice of gate library . in section 3we have described different architectures for flip flops . in section 4we have compared the circuit architectures proposed in the present work with the existing circuit architectures for the same .finally we conclude the work in section 5 .to provide a systematic protocol for designing reversible sequential circuit and to compare the proposed circuit architectures with the existing architectures we need to address certain conceptual issues related to reversible circuit designing . to be precise , conceptual issues related to feedback , choices of gate library and approximate optimization ( local optimization ) techniques will be addressed in the following subsections .   .it is widely believed that feedback is not allowed in a reversible circuit [ [ vasudevan ] ] .this is true if we consider feedback in a similar fashion as it is dealt in classical irreversible logic .the objection against feedback is twofold .firstly , merging of two computational paths is not allowed in a reversible circuit and secondly , time axis goes from left to right in a reversible circuit ( as shown in fig .thus if we need to follow the same notion of time axis in a reversible sequential circuit then feedback will essentially mean a journey in negative time axis or existence of time machine .this is against the notion of physical reality . but these strong objections against feedback in reversible circuit can be circumvented by establishing the equivalence between the circuits in fig .1a and fig . 1b . to be precise , the feedback loop shown in fig .1b is only in space not in time .therefore , the circuit in fig .1b is equivalent to a cascaded circuit in time axis ( see fig .thus the usual notion of time axis is not valid in reversible sequential circuit ( i.e. in a circuit having spatial feedback loop similar to one shown in fig .further , since the circuit in fig .1b is equivalent to the cascade shown in fig 1a , there is no merging of computational paths and consequently there would not be any loss of energy provided u is unitary .this conclusion coincides with the toffoli s idea [ [ toffoli ] ] of unitary transition function .now if we follow , this notion of feedback , then to establish the reversibility of the architecture it would be sufficient to establish the unitarity of u. here we would also like to note that in this restricted notion of spatial feedback we can not allow any arbitrary feedback loop .an allowed loop has to be reducible to structure shown in fig .1b .   in the previous subsectionwe have shown that in order to design a reversible sequential circuit we have to design u ( in fig .1b ) as unitary .now if we know the truth table of u and wish to decompose u in terms of finite number of logic gates , we can use one of the two existing approaches . in the first approach ,one designs reversible gates equivalent to irreversible logic gates .for example , see fig . 2 in which the classical irreversible gates like nand , and , nor etc are replaced by corresponding reversible gates for reversible circuit , constructed by combination of not , cnot and ccnot gates .in this approach after designing the equivalent gates one can substitute each irreversible gate of a conventional circuit by corresponding equivalent reversible gates and obtain the required reversible circuit .this straightforward approach is used in earlier works [ [ picton]-[hassan babu ] ] but the application of this approach is limited since it requires an existing reversible circuit and it can not go beyond the limits of classical computation . herewe would like to note that all earlier efforts of designing sequential reversible circuits [ [ picton]-[thapiyal2 ] ] were limited to this approach .for example , j. e. rice [ [ rice ] ] has substituted nor gate ( present in a conventional irreversible circuit ) by ccnot and has used fredkin gate to generate fanout from the clock .h. thapiyal [ [ thapiyal2 ] ] has substituted nand gate by new gate and and gate by fredkin gate . in the second approach onestarts with a truth table ( defining the desired logic ) and the gate library and then the corresponding reversible circuits are designed by transforming the inputs into outputs .unidirectional and bidirectional synthesis algorithms [ [ maslov1 ] ] are example of this approach . in principlewe have followed this approach , but instead of following an existing synthesis algorithm we have developed the designs by utilizing the logical symmetry of the truth table .for example , a careful look on the truth table of conventional sr flip flop tells us that there is an unstable condition and as is complement of , so both of them can never take same values .let us consider two cases : ( i ) when clock is high , and the last state is in set condition and ( ii ) when clock is high and the last state is in reset condition . nowif we operate the flip flop in set condition ( i.e. and ) then the state obtained in the output in both of the above cases will be and .since we obtain same result for two different cases , it always violates bijectivity .this problem may arise in the circuits designed by the first approach and this fact is reflected in the state table of sr latch reported by rice [ [ rice ] ] .the truth table provided by rice ( see table v in [ [ rice ] ] ) is not bijective as it gives two different inputs ( 0100 and 0101 ) .further we would like to note that number of feedback loops present in an irreversible sequential circuit can not be reduced , if we adhere to the first approach .whichever synthesis algorithm we follow , it is important to define a gate library . the definition of the gate library ( i.e. gates which are the member of that library ) is not well defined and there does not exist any single convention . the physical complexity of gates may not be same in two different implementation of quantum circuits .for example , it may be easy to build an arbitrary gate a in nmr technology but it may not be that easy in superconductivity based technology .a n - qubit quantum gate is represented by  unitary matrix and product of any arbitrary number of unitary matrices is unitary .consequently , if we put a set of quantum gates in a black box then an unitary matrix will represent the box and one can technically consider it as a new gate .if we allow such construction of new gates then any circuit block ( of arbitrary size ) can be reduced to a single new gate , provided it does not contain any measurement operation .thus it is straightforward to observe that the use of new gate to reduce the gate count ( as it is done in [ [ thapiyal2]-[hassan babu ] ] ) is an artifact . to be precise , we would like to mention that h. thapiyal _et al _ [ [ thapiyal2]-[thapiyal4 ] ] has introduced new gate , tks gate and tsg gate to reduce the gate complexity of several circuits .similarly , h. m. hasan babu ( see fig . 7 in [ [ hassan babu ] ] ) has introduced another new gate to reduce the gate count of a full adder circuit .thus the gate count ( gate complexity ) reported in these works [ [ thapiyal2]-[thapiyal4 ] ] are misleading and consequently we need a logical approach to construct a gate library which in turn will help us to compare more than one circuit designs proposed for the same purpose .  all the 1 qubit gates ( set of all phase gates ) along with any two qubit gate forms an universal set of quantum gates .since cnot is a two qubit gate , which have been experimentally realized by different groups by using different techniques , it seems logical to construct a gate library whose elements are phase gates and cnot in case of quantum circuit and cnots in case of classical reversible circuits .this is why we have chosen a gate library which contains not ( n ) , cnot ( c ) and toffoli ( t ) as its element .this particular choice of gate library ( nct ) is not only logical but also consistent with the existing approaches [ [ maslov1],[ad vos ] ] . since phase gates and cnot forms a set of universal gate, we can construct toffoli with phase gates and cnot and that requires five gates .we have used these facts to compare the gate complexity of our designs of sequential circuits with the existing proposals .the comparison is done with respect to the ( nct ) gate library ( as shown in table i ) and also with respect to the universal gate library , comprising of phase gates and cnot ( as shown in table ii ) .even if we start with the same truth table and same gate library and use the second approach of synthesis then also different logical paths may lead to different circuits .we have followed three different logical paths and have obtained three different architectures for every flip flop . after designing a circuitwe need to optimize it but an exact optimization technique s time complexity is [ [ anindita ] ] where , is the number of qubit line present in the circuit , is the total number gate present in the circuit and , is the number of qubit associated with the largest gate present in the gate library .thus in the present case when the gate library is ( nct ) then and the time complexity of exact optimization algorithm is this increases exponentially with and . in order to avoid this exponential rise in time ,certain approximate optimization ( local optimization ) algorithms ( e.g. bi - directional algorithm , template matching algorithm ) have been designed and in practice we use them . since the optimization algorithm is an approximate one , it may lead to different circuit architecture but the order of gate complexities have to be the same .this fact is clearly reflected in the table i and table ii given below .it is clear from the earlier approaches on reversible circuit designing that there is no unique circuit for a particular purpose and even if we start with the same gate library different synthesis technique may yield different circuit architecture . more importantly ,if we obtain two different circuit architectures for the same purpose ( which satisfies the same truth table ) by using two different synthesis techniques and then apply same approximate circuit optimization algorithms ( e.g. template matching algorithm , unidirectional algorithm , bi - directional algorithms etc . ) on them it is not essential that the final circuit will be same .we have used t for toffoli ( ccnot ) gate , f for feynman ( cnot ) gate andn for not gate .the sum of , and gates present in a circuit gives us the gate count or the gate complexity of the circuit .here we have used a different approach and ( nct ) gate library to design reversible memory elements/ flip flops . to be precisewe have obtained three circuits for sr flip flop , jk flip flop , d flip flop , t flip flop , master slave d flip flop , and master slave jk flip flop . here refers to first feedback based design , refers to second feedback based design and refers to design without feedback which , mainly uses refresh mechanism except in t flip flop .we have included all the circuits in the present work because of the fact that no particular design appeared as better than the other .this fact can be seen clearly from table i where we can see that the third design is best to construct t flip flop but the first design is better as far as construction of sr flip flop is concerned and all three approaches used by us yield better results compared to earlier works with respect to the most logical gate library ( nct ) . the resultant reversible flip flop eliminates the unstable condition found in sr flip flop and also minimizes the number of feedback loops in the jk flip flop and master slave jk flip flop .the logic of each of the circuit designed is discussed in the following subsections .   [ cols="^,^ " , ]   +    it is interesting to note that the advantages of our design over the earlier proposals and all other conclusions remained same .this approach of counting circuit complexity by considering complexity of toffoli as 5 is consistent with the earlier works [ [ maslov1 ] ] .in section 2 of the present work , we have addressed the conceptual issues related to the designing and optimization of reversible circuits in general with a special attention towards the issues related to the designing of reversible sequential circuits . in this sectionwe have shown that it is required to define an acceptable gate library and a good choice for that can gate library .further , it has been shown that the advantage in gate count obtained in some of the earlier proposals by introduction of is an and if it is allowed then every circuit block ( unless there is a measurement ) can be reduced to a single gate .the important conclusions of section 2 have been used in the next section to design circuits for sr flip flop , jk flip flop , d flip flop , t flip flop , master slave d flip flop and master slave jk flip flop . in section 4we have seen that our the unstable condition i.e. it always holds the last state but also uses lesser number feedback loops ( i.e. only 2 feedback loops against 4 in conventional jk flip flop ) . in the third design feedback loopsare replaced by application of refresh mechanism on some qubit lines which can also be reduced ( in sr flip flop it can be minimized to 1 refresh operation on qubit line ) but it leads to increase of gate count and consequently it has been avoided in the present work . from the comparison tables( table i and table ii ) , it is clear that the none of the earlier proposals have lesser gate complexity and lesser number of garbage bits compared to the present proposals .the sequential circuits described here uses only nct gates and there exist several proposals for realization of cnot and ccnot gates using cmos based technology [ [ vasudevan]-[ad vos ] ] .thus it is technically possible to construct the proposed reversible sequential circuits with the help of conventional cmos based technology .consequently , we can easily build classical reversible memory element .but the implementation of the present work is not limited to classical domain , this is because of the fact that we can also implement the proposed circuits in quantum domain with the help of nmr or quantum dot or optical implementation [ [ the : cerf - teleportation ] ] .if one aims to provide optimized reversible circuits for all the useful component of a classical computer and then this work along with the proposal of [ [ ad vos ] ] will help him to provide a complete architecture for a classical reversible computer . since it will be free from the problem of decoherence and scalability it seems more practical and easy to built than a real scalable quantum computer .[ maslov1]a transformation based algorithm for reversible logic synthesis , d. m. miller , d. maslov , g. w. duek , proceedings of design automation conference ( dac03 ) , anaheim , califomia ( 2003 ) 318 .[ thapiyal]reversible logic to crytographic hardware : a new paradigm , h. thapiyal , proceedings of the ieee international midwest symposium on circuits and systems ( mwscas 2006 ) puerto rico ( 2006 ) .[ thapiyal3]novel design and reversible logic synthesis of multiplexer based full adder and multiplier .h. m. thapiyal and m. b. srinivas , ieee international midwest symposium on circuits and systems , * 2 * ( 2005 ) 1593 .[ thapiyal4]a new reversible tsg gate and its applications for designing efficient adder circuit , h. m. thapiyal and m. b. srinivas , international symposium on representations and methodologies of future computing technologies ( rm2005 ) , tokyo , japan ( 2005 ) .

Abstract: circuits for sr flip flop , jk flip flop , d flip flop , t flip flop , master slave d flip flop and master slave jk flip flop have been provided with three different logical approaches . all the circuits have been optimized with the help of existing local optimization algorithms ( e.g. template matching , moving rule and deletion rule ) and the optimized sequential circuits have been compared with the earlier proposals for the same . it has been shown that the present proposals have lower gate complexities and lower number of garbage bits compared to the earlier proposals . it has also been shown that the advantage in gate count obtained in some of the earlier proposals by introduction of new gates is an and if it is allowed then every circuit block ( unless there is a measurement ) can be reduced to a single gate . further , it is shown that a reversible flip flop can be constructed even without a feedback . in this context , some important conceptual issues related to the designing and optimization of sequential reversible circuits have also been addressed .   * on the synthesis of sequential * * * * circuit *  anindita banerjee and anirban pathak  department of physics and material science and engineering , jiit university , a-10 , sector-62 , noida , up-201307 , india .


the standard clustering problem consists of partitioning a set of input vectors , such that the vectors in each partition ( cluster ) are close to one another according to some predefined distance function .this formulation is the objective of the popular -means algorithm ( see , for example , ) , where denotes the final number of clusters and the distance function is defined by the -norm .another similar example of this formulation is the -median algorithm ( see , for example , ) , where the distance function is given by the -norm .clustering a set of input vectors is a well - known np - hard problem even for clusters .several approximation guarantees have been shown for this formulation of the standard clustering problem ( see and references therein ) .intensive recent research has focused on the discovery of homogeneous substructures in large matrices .this is also one of the goals in the problem of _biclustering_. given a set of rows in columns from a matrix , a biclustering algorithm identifies subsets of rows exhibiting similar behavior across a subset of columns , or vice versa .note that the optimal solution for this problem necessarily requires to cluster the vectors and the dimensions simultaneously , thus the name biclustering .each submatrix of , induced by a pair of row and column clusters , is typically referred to as a _bicluster_. see figure [ fig : ex ] for a simple toy example .the main challenge of a biclustering algorithm lies in the dependency between the row and column partitions , which makes it difficult to identify the optimal biclusters .a change in a row clustering affects the cost of the induced submatrices ( biclusters ) , and as a consequence , the column clustering may also need to be changed to improve the solution .  finding an optimal solution for the biclustering problem is np - hard .this observation follows directly from the reduction of the standard clustering problem ( known to be np - hard ) to the biclustering problem by fixing the number of clusters in columns to . to the best of our knowledge, no algorithm exists that can efficiently approximate biclustering with a proven approximation ratio .the goal of this paper is to propose such an approximation guarantee by means of a very simple scheme .our approach will consist of relieving the requirement for simultaneous clustering of rows and columns and instead perform them independently .in other words , our final biclusters will correspond to the submatrices of induced by pairs of row and columns clusters , found independently with a standard clustering algorithm .we sometimes refer to this standard clustering algorithm as one - way clustering .the simplicity of the solution alleviates us from the inconvenient dependency of rows and columns .more importantly , the solution obtained with this approach , despite not being optimal , allows for the study of approximation guarantees on the obtained biclusters . herewe prove that our solution achieves a worst - case approximation ratio of under -norm for 01 valued matrices , and of under -norm for real valued matrices .finally , note that our final solution is constructed on top of a standard clustering algorithm ( applied twice , once in row vectors and the other in column vectors ) and therefore , it is necessary to multiply our ratio with the approximation ratio achieved by the used standard clustering algorithm ( such as ) . for clarity, we will lift this restriction in the following proofs by assuming that the applied one - way clustering algorithm provides directly an optimal solution to the standard clustering problem .this basic algorithmic problem and several variations were initially presented in with the name of direct clustering .the same problem and its variations have also been referred to as two - way clustering , co - clustering or subspace clustering . in practice , finding highly homogeneous biclusters has important applications in biological data analysis ( see for review and references ) , where a bicluster may , for example , correspond to an activation pattern common to a group of genes only under specific experimental conditions .an alternative definition of the basic biclustering problem described in the introduction consists on finding the maximal bicluster in a given matrix .a well - known connection of this alternative formulation is its reduction to the problem of finding a biclique in a bipartite graph  .algorithms for detecting bicliques enumerate them in the graph by using the monotonicity property that a subset of a biclique is also a biclique  .these algorithms usually have a high order of complexity .we assume given a matrix of size , and integers and , which define the number of clusters partitioning rows and columns , respectively .the goal is to approximate the optimal biclustering of by means of a one - way row clustering into clusters and a one - way column clustering into clusters .  for any denote = \{1,\ldots , t\} ] and  ] and  ] , and a special column norm , ))} ] into clusters such that the cost function is minimized .analogously , the one - way clustering of columns  ] , \setminus o_r , o_c) ] , and .note that , , and are simply blocks of bicluster , which we need to make explicit in our notation for the proof .  changing a 0 to 1 in ora 1 to 0 in decreases by two , while changing a 0 to 1 or 1 to 0 in or changes by at most one .it follows that _ swapping _ a 1 in or with a 0 in ( see figure [ fig : swaps]a ) , or swapping a 1 in with a 0 in , or ( see figure [ fig : swaps]b ) decreases while remains unchanged . in other words , in a solution that minimizes no such swaps can be made . in the remainder of this subsection , we assume that the bicluster satisfies this mentioned property .it follows that ( i ) , and are blocks of 1 s , ( ii ) is a block of 1 s and is a block of 0 s , or ( iii ) , and are blocks of 0 s .denote by the number of 1 s in a given block .it follows that , and .we denote , , , , and and rewrite equation ( [ eq : thbound ] ) as with constraints  ]  ] ; ( ii ) ,  ] and ; or ( iii )  ] . hence , equation ( [ eq : thbound ] ) is satisfied for -norm and real valued matrices when .we have shown that approximating the optimal biclustering with independent row- and column - wise standard clusterings achieves a good approximation guarantee .however in practice , standard one - way clustering algorithms ( such as -means or -median ) are also approximate , and therefore , it is necessary to multiply our ratio with the approximation ratio achieved by the standard clustering algorithm ( such as presented in ) to obtain the true approximation ratio of our scheme .still , our contribution shows that in many practical applications of biclustering , it may be sufficient to use a more straightforward standard clustering of rows and columns instead of applying heuristic algorithms without performance guarantees .we thank nikolaj tatti for reading through the manuscript and giving useful comments .

Abstract: the problem of biclustering consists of the simultaneous clustering of rows and columns of a matrix such that each of the submatrices induced by a pair of row and column clusters is as uniform as possible . in this paper we approximate the optimal biclustering by applying one - way clustering algorithms independently on the rows and on the columns of the input matrix . we show that such a solution yields a worst - case approximation ratio of under -norm for 01 valued matrices , and of under -norm for real valued matrices .


entering a new social group , we are vividly interested in all kinds of non - formal contacts .they are necessary to interpret and qualify properly all information we get : as relevant or marginal , unique or commonly available , urgent or not so , etc .we are taught by evolutionary psychology that this need reflects the way of work of the human brain , as it has been formed during millions of years of evolution .this need forms then our today s relations with people as well . as a consequence ,it remains relevant for any social theory of human relations .that is why gossip appeared as an appealing catchword .because of its roots noted above , theory of gossip can be seen as a part of evolutionary psychology .once an evolutionary sociology emerges , we will certainly find it there .  in sociophysics , we look at social sciences through a mathematical glass .being somewhat blind to hermeneutical analyzes , we look for determinism , structure and numbers .such an attitude meets an old hope of sociologists to deal with problems as well - defined and narrow as physicists have .( invocations to physics are quite frequent in old sociological textbooks , to call only few examples . )sociology can meet with sociophysics in all cases where the structure of society is of importance . by structure we mean a system of mutual or directed connections between people . in a reductionistic approach , sucha system can be represented by a graph , where people are nodes and relations between people are links . quite naturally , such a picture is a favorite tool in sociophysics .here we are going to use this mathematical representation to analyze the spreading of gossip .the starting point is the theory and experiment proposed and performed recently by some of present authors . according to this theory , `` gossip is non - public information about knowable people and its primary attribute is proliferation .gossiping is a communicative propensity characteristic of the human race manifesting itself in smaller communities '' .then , the person who is the subject of the gossip is known personally to the community .this fact makes the gossip interesting and this interest is the necessary condition of the gossip spreading .this `` semiprivate '' character makes our case different from the theory of rumor by galam . the experiment dealt with an interest in gossip about a known or knowable person in a web - based social network . as a result, three levels of social inclusion have been found , which practically limited the gossip spreading . as a sample of the questionnaire ,answers were gathered to the following :  1 .would you tell about your girlfriend s new job to your friend ? 2 .would you tell about your girlfriend s new job to your friend s girlfriend ?would you tell about your girlfriend s new job to your friend s girlfriend s colleague ?  the percentage of positive answers varied from through till , respectively for questions 1 , 2 and 3 .after the third degree the results had shown a sharp decline .these considerations led us to our main question , under what conditions a given gossip will be known in the whole community ?the above numbers ( ) served us as probabilities , that the gossip will be told to people of 1-st , 2-nd and 3-rd level of inclusion , defined by the questionnaire . at this pointwe are faced with the as - yet - unsolved problem , what is the structure of the social network ?we have to admit that the answer varies from one kind of network to another , one or another kind of social ties . in the literature of the subject, one can find arguments about different parameters of social networks : size from a hundred to three hundreds and more , clustering coefficient , strength of ties and structure .the results can depend also on whether complete networks or personal networks are investigated .when we speak on friendly personal networks , the size of a typical group can decrease by at least one order of magnitude .not entering these discussions , here we attack the problem of gossip spreading in a model way , where the average number of friends is a model parameter .also , for simplicity we choose the random graph of erds and rnyi as a model of a social network .this selection should serve as a useful point of reference .the goal of this paper is to calculate the probability that the gossip is known , averaged over the community members .basically , the result is close to zero or one , except some range of the average number of friendship ties .this range can be seen as the range of a transition between two phases : `` they do not know '' and `` they know '' .however , even if the width of this range eventually shrinks to zero in the limit of large networks , this limit is not relevant for social sciences , where the quality of useful approximations does not necessarily increase with the system size .  in next sectionwe describe the model calculations and the results .last section is devoted to their discussion .from noted above , the following model assumptions emerge :  1 .  the set of nodes are those who `` know about '' , and that is why they are willing to hear .the links join two nodes if they are friends .the linkage is random , as in the erds rnyi model .4 .  the question is to evaluate the size of the group who will know the information .the detail is if the victim of the gossip is also a member of the network in which gossip is spread .the argument for this assumption is in the questionnaire `` about your girlfriend s new job '' . in this casethe talker role is to be limited to the set of boyfriends of the girl .however , we assume that the girl can have more boyfriends , and then the number of talkers can be greater than one .  in this casewe have two parameters : ( the number of nodes ) and ( the probability of a link of friendship between randomly selected nodes ) . as we know from the theory of random networks , the mean degree is .the numbers , and can be interpreted as weights in the average level of being informed about a given gossip .all friends of the girl who got a job will know it with probability one ( ) .this is a contribution .their friends ( each has still not informed ) will know the gossip with probability .this is a contribution .finally , consider friends of the friends ( supposed they are not informed yet ) .if each friend has uninformed friends , the information will pass to them from the teller with , say , probability .then , total level of being informed on the gossip would be this is a function of and .this expression has some deficiency : in the random networks the probability that two `` friends '' of a node are also `` friends '' is . in the above calculation, we disregarded this possibility .now we are going to include it . in the first zone , friends are informed with probability .each has neighbors , of them are already informed .then , newly informed are only next neighbors , and their contribution will be .how many still non - informed neighbors have these people ?the answer is that each has . they will be informed by a teller with probability .then , their contribution is .the total formula is : we note that still there are some assumptions left about the lack of correlations of further order , the arguments are somewhat heuristic , and valid only for small .however , obviously increases with . in the range where the formula is not valid ( large ) we rely on a computer simulation .  for given victim of the gossip ( one of nodes constituting network ) and all of its nearest neighbors ( talkers )we evaluate the number [ of paths of length [ from all other nodes to talkers . the probability that -th node _ is not _ informed is then , the level of being informed on the gossip for given victim is where summation goes over all and is not a talker .the total level of being informed of the gossip is averaged over all possible victims of the gossip in the given network we carry out our simulation for the set of probabilities , and .it appears ( fig .[ fig - xp ] ) that at some value of , almost everybody will know the gossip .\(a )  average probability of being informed on a gossip against ( a ) the probability and ( b ) mean node degree .theoretical curves ( eqs . and ) differ only slightly for , but coincide for larger networks .their accordance with the simulation results improves for larger , where the correlations between informed neighbors can be neglected with better accuracy ., title="fig : " ] + ( b )  average probability of being informed on a gossip against ( a ) the probability and ( b ) mean node degree .theoretical curves ( eqs . and ) differ only slightly for , but coincide for larger networks .their accordance with the simulation results improves for larger , where the correlations between informed neighbors can be neglected with better accuracy ., title="fig : " ] +  this value of is however not strictly defined and it depends on the system size . for small ,both expressions ( eqs . and ) , for correlated and uncorrelated ( i.e. random ) case , work almost equally well .motivated by tradition of statistical mechanics , we made an attempt to evaluate the probability , where .this can be seen as a critical value between the two phases remarked above , where `` they do not know '' for and `` they know '' for .the size dependence of , i.e. , is shown in fig .[ fig - pcn ] .the results nicely fit a power law .the exponent slightly varies with the measured probabilities ( , and ) ; it is 0.68 for the values of the probabilities ( 1.0 , 0.748 and 0.221 ) used here , but 0.63 for ( 1.0 , 0.7 and 0.25 ) , 0.65 for ( 1.0 , 0.75 and 0.25 ) and 0.66 for ( 1.0 , 0.8 and 0.2 ) .we made also an attempt to evaluate the influence of the clustering coefficient on our results .the coefficient is defined as the ratio of number of links between nearest neighbors of -th site , divided by the maximal value of this number and averaged over all sites of the network with more than one neighbor .our motivation comes from the suggestion that in social systems , the correlation is larger than for random case .the simulation is performed for and , which is equal to for the random ( i.e. uncorrelated ) network .the clustering coefficient is increased by a rewiring procedure : a node is selected with at least neighbors , and the link to one of its neighbor is cut ; instead , it is added between two remaining neighbors .the result is that as increases , the average size of informed group decreases .example of this result is shown in fig .[ fig - xcc ] .dependence of critical probability on the system size .the solid line shows the least square fit with . ]it is clear that in the case of larger , information is transmitted more frequently within a small group . on the contrary, its spread over the whole community is less effective .this effect is parallel to the discussion in sociological literature , where links joining different compact groups ( the so - called weak ties ) are considered to be crucial for the information spreading .influence of the clustering coefficient on the level of gossip propagation for .increasing and keeping constant , we eventually get the network split in parts . ]when a social group is formed from the beginning , almost nobody knows anything about others . soon mutual ties are built and strengthen , and information starts to flow . in our picture , this process can be interpreted as an increasing of the probability in time .the results presented in fig . 1 indicate , that the information carried by gossips increases initially with as a low degree polynomial .gradually , the whole group becomes informed .  keeping the experimental values of constant , as we do, we can expect some characteristic distance from the victim to a member who is informed with probability , say , 1/2 .( this distance is a graph characteristics and it should not me mixed with the social distance , discussed elsewhere ) .certainly , this distance depends on the numbers , . on the other hand ,the diameter of the random network can be evaluated as . at can expect that . approximating by , we get , i.e. comparing this with our numerical result , we get close to . having in mind our values of the probabilities ,we are not surprized with this distance .reasonably enough , it agrees with the interpretation of the experiment , given in our previous work entitled _ three levels of inclusion _ .we conclude that the exponent is not universal , but it depends on the probabilities . with increasing , the whole group is gradually dragged into the shell of radius around the victim. actually , the gossiping can be a good reason to enhance group ties .  in statistical mechanics, our results may be relevant for the percolation problem in random networks .it is known that large connected clusters appear for  .important difference is that in our case of gossip , we have one source of information . in this aspect , the gossip spread can be compared to a spread of infection , e.g. in networks of sexual interactions .once we allow for a distribution of sources within the network , the problem of gossip becomes alike to the family of problems , as bootstrap percolation or diffusion percolation .it seems natural that these problems will find social applications , similar to those of standard percolation theory .the bottleneck here is the sociological experiment , which is much more difficult , costful and debatable than computer simulations .three of the authors ( z.s , b.s . and k.k .) are grateful to the organizers of the eighth granada lectures in 2005 , where this work was initialized , for their kind hospitality .calculations were carried out in ack cyfronet the machine time on hp integrity superdome is financed by the polish ministry of science and education under grant no .mnii / hp_i_sd / agh/047/2004 .although the number of a search results in google can hardly serve as a measure of reality , we note that it shows web pages with `` evolutionary psychology '' and only 1220 web pages with `` evolutionary sociology '' . for a comparison , `` physics '' reveals web pages .these numbers suggest differences of scale .m. s. granovetter , _ am .j. sociol . _* 78 * ( 1973 ) 1360 d. j. watts , p. s. dodds , m. e. j. newman , _ science _ * 296 * ( 2002 ) 1302 j. m. pujol , a. flache , j. delgado , r. sangesa , _ j. artificial soc .( 2005 ) http://jasss.soc.surrey.ac.uk/8/4/12.html

Abstract: we consider the average probability of being informed on a gossip in a given social network . the network is modeled within the random graph theory of erds and rnyi . in this theory , a network is characterized by two parameters : the size and the link probability . our experimental data suggest three levels of social inclusion of friendship . the critical value , for which half of agents are informed , scales with the system size as with . computer simulations show that the probability varies with as a sigmoidal curve . influence of the correlations between neighbors is also evaluated : with increasing clustering coefficient , decreases .


he temperature - emissivity separation ( tes ) problem bedevils any attempt to extract spectral information from remote sensing in the thermal infrared .a variety of methods has been proposed for handling the temperature - emissivity separation ( tes ) problem  . in most of them , simultaneouslst and band emissivity retrieval depends upon specifying an emissivity value in one or more reference bands .the modis land surface temperature ( lst ) algorithm  seeks a pair of reference channels in a part of the thermal spectrum in which the emissivity of natural surfaces displays very limited variation , and may therefore be regarded as known with good confidence .multiband emissvities inferred on this basis are called `` relative '' emissivities  .other algorithms of this nature include the reference channel method  , emissivity normalization  , temperature - independent spectral index method  ,  and spectral ratios  .the study by li et al . shows that all of these relative emissivity retrieval algorithms are closely related , and argues that they may be expected to show comparable performance .the analysis of multispectral thermal imager ( mti ) data  depends on collection of radiance from a surface with looks at nadir and 60 degrees off - nadir , assuming a known angular dependence of emissivity , in order to balance equations and unknowns .the generalized split - window lst algorithm  likewise uses dual looks in a regression - law based approach . the `` grey body emissivity '' approach  exploits the slow variation of emissivity with wavelength for certain natural targets , while the physics - based modis lst algorithm  exploits observations taken at day and at night , on the assumption that band emissivites do not change over periods of a few weeks .a study with the airborne hyperspectral scanner compares multiple tes approaches .we shall investigate a simulated annealing approach to the tes problem .the approach is an extension of earlier work on bayesian tes .simulated annealing can not give a unique solution to this problem , but we shall prove that the average of a large number of simulated annealing tes solutions converges almost surely to a unique tes estimate .this paper will concentrate on the mathematical basis of the algorithm and a proof of its convergence .a study of its performance will form the subject of a subsequent paper .simulated annealing has traditionally been regarded as a preferred method of global solution for combinatorial optimization problems such as traveling salesman . in this paper , we adapt the metropolis algorithm to an optimization problem that lacks a unique global optimal solution : temperature - emissivity separation .the underdetermined temperature - emissivity separation ( tes ) problem , notoriously - , has a continuous infinity of solutions that yield the identical optimum value for any cost or payoff function one cares to choose .a key part of any simulated annealing algorithm is the choice of an annealing schedule that causes the posterior probabilities to transition from nearly uniform to very tight in such a way as to evade the risk of the map search from converging to a local , rather than a global , optimum .the look and feel of the justification for this approach is ergodic .  in what followswe shall mostly concern ourself with the existence of a solution to the simulated annealing tes problem , and shall simply assume that a suitable annealing schedule has been supplied .factors that enter into the choice of annealing schedule are described in .selection of the annealing schedule and sample tes retrievals will be the subject of a subsequent paper .suppose that we have in our possession prior knowledge that a target patch that forms part of the lower boundary of the atmosphere is composed of an intimate mixture of spectral endmembers at temperature .for later convenience , we shall require that spectral emissivities be bounded continuous functions of wavenumber with bounded first derivative with respect to wavenumber. the label may , depending upon context , refer to wavenumber , or to a finite number of wavenumber - averaged spectral bands . except in section [sect : arbitrary ] , we shall assume the band interpretation .the spectral mixture amounts to a mapping into a geometric m - simplex whose vertices have spectral endmembers at a stipulated temperature for coefficients .suppose we have distinct points in chosen so that the vectors are linearly independent .then the set with and is an m - simplex . a spectral mixture with stipulated weights corresponds to the vector  is to be interpreted as the fraction of the total surface area of the patch occupied by the -th subregion .the product is thus a normalized emissivity - area product for that subregion .we defer the case of endmembers with differing temperatures to a later date .we believe it , however , to be a straightforward extension of the reasoning in this paper . ]  the _ interior _ of is the subset of for which , that is the closure of its interior .the polyhedron of , denoted , is the set comprised of the points of considered as a subset of , and is a convex compact subset of .  in the case , a familiar example of a 2-simplex is the ternary diagram used to classify phreatic igneous rocks .the double three - component diagram used in the qapf classification  scheme is a union of two 2-simplices , and is an example of a simplicial complex . for present purposes ,the pure endmember for the trial is assigned to the vertex of  with the spectral mixture corresponding to a point in the polyhedron of ,  it is necessary to account for surface temperature in a somewhat different way .let the minimum and maximum physically admissible surface temperatures be and , respectively .then the temperature of our target patch is given by with corresponding to introduced already , we have from eq .( [ eq : t_homotopy ] ) the unit interval , with the quantity that appears in the forward model for the trial is  is the ( band - integrated , as needed ) planck function at temperature .the parametrization of the choice in terms of the vector is a mapping into the topological product of and .the set is not a simplex , nor is it necessarily a simplicial complex .it is , however , a convex polytope , and is the convex hull of its vertices .can be decomposed into either a simplicial complex or a union of simplices . ]we score trial mixtures by that we most wish to maximize : the posterior probability for the observed spectral radiance to originate from a surface patch with temperature and spectral emissivity . a standard argument gives the posterior probability in terms of a maxent estimator \frac{di}{\sigma(t_{a } ) } \label{eqn6}\ ] ] in terms of a forward model that is some function of the trial , in each spectral bin .we note that while the equation of transfer is linear , the dependence of its solution upon need not be .the assumed noise variance is shown as having a formal dependence upon a parameter , the `` annealing temperature '' , which governs the annealing schedule for the search for a maximum a - posteriori solution .the joint posterior probability in spectral bands is proportional to \frac{di}{\sigma(t_{a } ) } \label{eqn7}\ ] ]  if radiance in each of j bands originating from a patch on the earth s surface has been detected at the top of the atmosphere ( toa ) , the posterior probability that the surface patch is at a temperature t given prior knowledge is given by bayes theorem as the noise variance is assumed known and the functional dependence of probabilities upon is omitted .the prior probability for the radiances has no dependence upon and for our purposes may be absorbed into an overall normalization . equation ( [ eqn3 ] ) is evaluated with aid of the prior probability for the surface to be at temperature t and have spectral emissivity , given available knowledge  ,  is the conditional probability for the hypothesis that the surface temperature is , and the spectral emissivity , given observed radiances and prior knowledge .each trial is thus scored according to the joint posterior probability for observed spectral radiance to result from surface temperature and spectral emissivity , where stands for .thus , in going from the to the trial , the candidate mixture is selected by monte carlo draw and for the new trial is compared to for the last one .the probability that it is accepted is  where the probability of taking a downward step in is determined by the annealing schedule .the dependence of on the annealing schedule is symbolized by the annealing `` temperature '' which is taken to decrease systematically during the map search .the actual form takes in practical calculations is determined empirically .we now examine the question of convergence . corresponding to the sequence of m - simplices as the number of trials increases without boundis a sequence of trials with associated loci .  asa closed bounded subset of , is a compactum .therefore , as , the sequence of trials contains a convergent subsequence , whatever the value of . correspondingly , the sequence of posterior probabilities likewise has a convergent subsequence that , by construction , tends to the maximum value of the posterior probability , i.e. , to a map solution for and .consider the map given by the mapping eq .( [ eq : nextmap ] ) gives the action of the metropolis algorithm according to eq .( [ eq : metropolis ] ) at sufficiently late times in the annealing schedule that a transition to a state of decreased posterior probability occurs rarely ; in the limit , almost never . we have noted that at a sufficiently late point in the annealing schedule , trials that decrease the posterior probability eq .( [ eq : posterior ] ) will become infrequent .we may elide any such trials without affecting the convergence of the subsequence , which then takes the form for all greater than some , convergence of the subsequence implies the cauchy condition ( with the euclidean norm supplying a suitable metric for finite ) so that the mapping eq .( [ eq : nextmap ] ) generates a sequence of trials for which is nondecreasing .by zorn s lemma , the set comprised of all admissible trials has at least one element with a maximal value of .we note that maximizing also maximizes the information - theoretic entropy by eq .( [ eqn6 ] ) . according to the usual statement of the second law ,the state of maximum entropy is one of thermodynamic equillibrium .we may therefore , in a nod to refs . and ( both , call the limit eq . ( [ eq : phifp ] ) an equillibrium point .we note that , in the limit , eq . ( [ eq : phifp ] ) amounts to a fixed point of eq .( [ eq : nextmap ] ) .ordinary fixed - point theorems are inapplicable to eq .( [ eq : nextmap ] ) because it is neither continuous nor semicontinuous : it can map an open set to a singleton .we can , however , adapt a celebrated construction introduced by nash  to prove the existence of a fixed point of an equivalent self - mapping .in fact , we shall prove a somewhat stronger result .consider for stipulated .the function is continuous in the mixture .define the mapping by where the index is taken to run over members of any finite set of admissible trials in the execution of the metropolis algorithm .( one may think of the collection of all sequences in ensemble - theoretic terms . )suppose that is a fixed point under eq .( [ eq : nashmap ] ) . in eq .( [ eq : nashmap ] ) some values of correspond to choices for for which the posterior probability does not increase : for these values of , if the choice is fixed under the mapping in eq .( [ eq : nashmap ] ) , then the contribution to from any must not decrease ; therefore , , lest the denominator in exceed unity .put another way , no other choice of can increase the posterior probability .but that is the definition of an equillibrium point .if , on the other hand , an equillibrium point maximizes the posterior probability eq .( [ eq : posterior ] ) , every vanishes , so that is a fixed point .equation ( [ eq : nashmap ] ) is continuous and maps points into a convex compactum .a fixed point exists according to the brouwer fixed - point theorem that , by construction , maximizes the _ a - posteriori _ probability of .the mapping eq .( [ eq : nextmap ] ) generates a sequence of trials for which is nondecreasing and gives the maximal value of in the limit , while eq .( [ eq : nashfp ] ) demonstrates the existence of a trial for which can not be made greater . in view of the ensemble - theoretic freedom to choose , we may identify the limit in eq .( [ eq : phifp ] ) with the fixed point in eq .( [ eq : nashfp ] ) .therefore , a convergent subsequence of annealing trials exists that tends to an equilibrium point .moreover , eq . ( [ eq : nashfp ] ) demonstrates that the annealing search can , in principle , find in a finite number of trials .we conclude that , granted a suitable annealing schedule , there exists at least one convergent sequence of trials that tends to map surface temperature and spectral emissivity estimates consistent with observed spectral radiances .the search algorithm just described assumes that the emissivity is describable by a mixture of a finite set of spectral end members .while the spectral mixture characterization of is of interest in its own right , it may be considered a stronger hypothesis than is strictly desirable . in particular, it seems intuitively reasonable that the simulated annealing approach to tes should work just as well - and admit a simpler algorithmic realization - by using trials with randomly chosen spectral emissivities , rather than by seeking a spectral mixture from a predetermined set of endmembers .in fact , it is possible to reduce the case of search using arbitrary for trials to an extension of the analysis in the preceding section by allowing the number of endmembers to grow without limit for each trial . instead of self - mappings into a single polytope with fixed , we consider a sequence of as , for each in the annealing schedule .the connection between arbitrary and a spectral mixture with whose endmembers are allowed to grow without limit is easily seen . if we chose from a set of randomly chosen endmembers , it is clear that any random can be constructed as a spectral mixture of other random endmembers . by induction: a single endmember trivially reproduces an arbitrary if it is chosen so .suppose that any equals a spectral mixture of suitably chosen random .then , by the inductive hypothesis for , it is possible to replicate any other by a mixture of some spectral emissivity of endmembers and a random endmember .we proceed by constructing the polytope for an arbitrary number of spectral endmembers .the polyhedron of the m - simplex with unit diameter may be circumscribed by an m - sphere of radius   is thus a subset of the topological product of n replicas of the unit interval  ]  \otimes  [ 0,1 ] \cdots,\ ] ] as is every convex polytope all the polyhedra and polytopes are compact , and by tychonoff s theorem , the set which circumscribes every and is likewise sequentially compact . is homeomorphic to the _ hilbert cube_. the hilbert cube \otimes  [ 0,\frac{1}{2 } ] \cdots  [ 0,\frac{1}{n } ]  \cdots\ ] ] is a subset of a hilbert space with the norm .however , we may still take the euclidean norm when choosing a metric . ] therefore a _ complete _ space : the sequence of trials , by sequential compactness of the hilbert cube , and sets homeomorphic to it , possesses a convergent subsequence whose limit is , by completeness of , an element of that space .the limit of the convergent subsequence of m - polyhedra is likewise contained within .the treatment of convergence in section [ sect : finitecase ] requires modification when the number of spectral endmembers is allowed to grow without limit . at each , the self - mapping and function applied to as before to give existence of an equilibrium fixed point .the simplex dimension is allowed to grow without bound , however . the resulting sequence of equilibrium points possesses a convergent subsequence for each value of .  with that caveat , as the sequence converges to a map estimate of and an arbitrary by the same reasoning used in the previous section .in consequence , the limiting equilibrium point of the convergent sequence as will tend to an estimate of the map value for for any admissible . we conclude that a suitable simulated annealing search will converge to an arbitrary spectral emissivity that gives a map estimate of .whatever the dimensionality of the spectral endmember parameterization of emissivity , sequential compactness guarantees existence of a convergent subsequence of trials . in practice, we must expect that there will be more than one such sequence .the nonuniqueness of solutions to the tes problem suggests that there will be a continuous infinity of possible trials that yield any stipulated value for the posterior probability . in any realizable search strategy , however , we need only contend with a countable set of convergent subsequences . amongst thesethere will be one for which the posterior probability is greatest .this will be the closest approach to the maximum a - posteriori solution achieved by simulated annealing . in the nature of things, more than one convergent subsequence may be expected to exist that yields this same maximal estimate , with the same asymptotic annealing temperature .we ignore all subsequences except these maximal ones .  in references and expectation values for and over the the posterior probability eqn .( [ eqn3 ] ) were shown to give good estimates for physical surface temperatures and emissivities .we claim that the mean of a large number of subsequences that converge to the limiting map value will tend to the expectation values for and with respect to eqn .( [ eqn3 ] ) .the maxent estimator is constructed from the posterior probability of noise power in a spectral bin . for the sake of simplicitywe assume identical noise power in each bin .a fully annealed map estimate may be thought of as an individual bernoulli trial drawn from the likelihood function for . by construction ,all such trials are independent and identically distributed with bounded expectation values . )are bounded despite bad behavior of the jeffreys prior at t=0 , because of the rapid decay of the exponentials away from the map solution , as rule demonstrates .] let and be the means of map surface temperature and spectral emissivity taken over over convergent subsequences .suppose the covariance matrix of the trials to be nonsingular .we invoke the multivariate central limit theorem to conclude the mean values converge weakly to the multivariate gaussian distribution : reliance on the mixing hypothesis in the form given by eq .( [ eq : mixed ] ) , however , brings with it the concern that the relevant covariance matrix might be singular . in that event ,the strong law of large numbers ensures and but without giving estimated variances of the mean values , such as come with eq .( [ eq : clt ] ) .  to the extent that the estimator used in the simulated annealing search is zero - mean error, we conclude the estimates yield accurate values for the physical values of and .as the spectral weights , lying as they do between zero and unity , possess bounded moments , this conclusion applies to the limiting mean values of as well .it is worth considering briefly the limiting case of infinite spectral resolution for . in this sectiononly , refers to wavenumber .we shall admit as endmembers any continuous function on a compact interval \in \mathbf{r} ] .thus , any physically admissible spectral emissivity may be approximated arbitrarily well by a suitable spectral mixture of an unlimited number of end members .we may regard the spectral mixture as an upper function on $ ] and take for a norm the lebesgue measure in the limit , with associated metric a mixture then becomes a vector in a banach space . as the sequence of limiting fixed points underthe self - mapping has a convergent subsequence which , again , satisfies the cauchy condition the argument from the cauchy property of the convergent subsequence of to the conclusion that the subsequence converges to a map equilibrium point likewise follows much as before . as  so that the convergent subsequence of annealing trials , again , tends to an equilibrium point .in section [ sect : convergence ] we proved convergence of simulated annealing searches for candidate map tes solutions . in section [ uniqueness ]we argued that the average of a large number of these candidate map solutions converges almost surely to a unique estimate of surface temperature and spectral emissivity that , given a forward model leading to an unbiassed estimator for and , closely approximates the true values of these quantities .  the motivation for seeking the nash equilibrium analogy came from the realization that spectral mixing theory amounts to the use of mappings into a simplex , and that iterative choice of weights in the tes problem according to an annealing schedule amounts to a self - mapping into a convex polytope .the analogy with spectral mixing theory , however , is incomplete : spectral unmixing , in either its reflective or thermal variants , generally appends and extra end member called `` virtual dark '' or `` virtual cold '' , used to accommodate the effect of contamination from noise and sensor artifacts .one may see the value of an extra end member by recalling that the highest order components in a principle components decomposition of a multivariate dataset tends to be dominated by noise and artifacts that do not correlate with the physical content of lower - order components .the role of a virtual garbage end member will be discussed in part two .a special case of great interest is the situation in which one seeks evidence that spectral radiance sensed at the top of the atmosphere ( toa ) contains evidence for the presence of a specific spectral component .this problem may be addressed by use of spectral mixtures comprised of the desired spectral component together with generic continuous end members whose number is allowed to grow without bound .the analysis of section [ sect : arbitrary ] , however , can not simply be modified by appending an endmember corresponding to the desired emissivity spectrum , as any completely random choice of may well be correlated with the stipulated spectrum and so introduce a spurious admixture of that endmember into the analysis . on the other hand , in a physical mixture of spectral endmemberssuch correlations might in fact occur naturally . without going into details ,we offer some thoughts on this problem .one way to proceed is to randomly select for the background in such a way that the otherwise arbitrary background emissivity spectra all lie in the null space of the stipulated endmember .the choice of background is thus made in much the same way as an empirical orthogonal basis set is selected .depending on the intended application , another way to handle this problem might be to find some way of marginalizing on the set of arbitrary background in the calculation of the posterior probability eq .( [ eq : posterior ] ) .if performed by the same sort of stochastic sampling typical of simulated annealing , this calculation would resemble a numerical approximation to the feynman - kac formula. ; in essence , a monte carlo path integral calculation .the practical utility of the mathematical development in this paper may be questioned .we address briefly two possible concerns .while convergence of the algorithm has been proved to our satisfaction , we have no equally satisfactory estimates of the rate of convergence , with the consequence that the choice of annealing schedule remains a matter of trial - and - error . in response to this concern, the availability of massively parallel computation made possible by the ready availability of cheap gpu arrays means that massive processing requirements need not preclude the use of a resource - hungry algorithm if that algorithm can provide performance not attainable by other approaches .part two of this study will address these issues .another legitimate concern is that the spectral emissivity of natural ground covers in the wild will seldom be known to the level of accuracy found in ref . . while true in general, this concern has not dissuaded other researchers from relying upon spectral unmixing .the simulated annealing approach to tes by spectral unmixing does , however , offer something that other tes algorithms do not : by construction , it gives ( in the limit ) the unique best estimate in a maximum _ a - posteriori _ sense , for the remote determination of surface temperature and spectral emissivity of a patch of ground that is known to be comprised of a spectral mixture of a stipulated set of spectral end members .dash , p. , f .-gttsche , f .- s .olesen , and h. fischer , `` land surface temperature and emissivity estimation from passive sensor data : theory and practice - current trends , '' _ int .j. remote sensing _ , vol .2563 - 2594 , 2002  li ,z - l , h. wu , n. wang , s. qiu , j. a. sobrino , z. wan , b - h tang , and g.yan , `` land surface emissivity retrieval from satellite data , '' international journal of remote sensing * 34 * , pp .3084 - 3127 , 2013     kealy , p. s. , and s. j. hook , `` separating temperature and emissivity in thermal infrared multispectral scanner data : implications for recovering land surface temperatures , '' _ ieee trans .remote sensing _ , vol .1155 - 1164 , 1993     borel , c. c. , and j. .szymanski , `` physics - based water and land temperature retrieval , '' in handbook of science algorithms for the multispectral thermal imager , b. w. smith , ed ., los alamos national laboratory and savannah river technology center , 1998   barducci , a. , and i. pippi , `` temperature and emissivity retrieval from remotely sensed images using the grey body emissivity method , '' _ ieee trans .remote sensing _ , vol .681 - 695 , 1996   sobrino , j. a. , j. c. jimnez - muoz , p. j. zareco - tejada .g. sepulcre - cant , eduardo de miguel , `` land surface temperature derived from airborne hyperspectral scanner thermal infrared data , '' _ rem .99 - 115 , 2006   morgan , j. a. , `` comparison of bayesian land surface temperature algorithm performance with terra modis observations '' , international journal of remote sensing * 32 * , 2012 , p. 8139 ; see also arxiv:0909.4461v3 , 2011       bretthorst , l. , `` bayesian spectrum analysis and parameter estimation , '' in berger , j. , s. fienberg , j. gani , k. krickenberg , and b. singer , eds , _ lecture notes in statistics _ , springer - verlag , new york , 1988 , vol .48  blumenthal , l. m. , and g. e. wahlin , `` on the spherical surface of smallest radius enclosing a bounded subset of n - dimensional euclidean space , '' bulletin of the american mathematical society * 47 * , pp .771 - 777 , 1941           john morgan is a senior engineering specialist in the sensing and exploitation department at the aerospace corporation , where his duties include studies of spacecraft sensor system performance and remote sensing phenomenology .he has a bs from caltech , and ms and ph.d .degrees from rice university , in physics .

Abstract: the method of simulated annealing is adapted to the temperature - emissivity separation ( tes ) problem . a patch of surface at the bottom of the atmosphere is assumed to be a greybody emitter with spectral emissivity describable by a mixture of spectral endmembers . we prove that a simulated annealing search conducted according to a suitable schedule converges to a solution maximizing the _ a - posteriori _ probability that spectral radiance detected at the top of the atmosphere originates from a patch with stipulated and . any such solution will be nonunique . the average of a large number of simulated annealing solutions , however , converges almost surely to a unique maximum a - posteriori solution for and .   the limitation to a stipulated set of endmember emissivities may be relaxed by allowing the number of endmembers to grow without bound , and to be generic continuous functions of wavenumber with bounded first derivatives with respect to wavenumber .   remote sensing ; temperature - emissivity separation ; surface temperature estimation .