proteins are intrinsically dynamic machines . their function , _e.g. _ the ability to respond allosterically to a a local perturbation such as ligand binding or chemical modification  , is intimately related to specific correlated vibrations , whose pattern is rooted in their native structure  . although many experimental techniques lend considerable insight in protein dynamics  , nmr spectroscopy is emerging as an increasingly powerful method for the characterization of correlated vibrational patterns involved in protein function and , notably , in allosteric intramolecular communication  .+ a large body of work in the paste decades has contributed to highlight the centrality of proteins three - dimensional folds to their function .protein scaffolds , for example , encode how energy flows across specific pathways encompassing certain _ hot - spot _ residues at key locations  , and linking different functional regions , but also sometimes portions of structure not apparently associated with function  .as it is well known , low - frequency , collective normal modes ( nm ) provide key fold - encoded patterns for conformational changes associated with functional motions , very often in excellent agreement with observed conformational changes  .however , typically these modes are highly damped due to the substantial exposure to the solvent of large portions of the protein structure  , suggesting that other , more localized and therefore more robust higher - frequency vibrations might be implied in protein function .+ the hypothesis that protein functional dynamics may imply the concerted action of large - scale motions sustained by _specific _ higher - frequency localized vibrations , although not new  , has been recently put forward  in the context of allostery , providing an intriguing rationale for the general idea that both slow and fast modes in proteins are connected to function  . as already argued in the early 70s by c. mcclare , for example , enzyme functioning may well imply non - thermal storage of energy in specific , fold - rooted localized vibrational modes , so as to lower the free - energy barriers of chemical reactions _ where it is needed _indeed , many experiments have affirmed the crucial role of specific localized vibrations , such as hinge motions , in mediating between faster atomic fluctuations and slower functional rearrangements  . + along the same lines, many experimental studies suggest that unusually long - lived vibrational modes may be excited in proteins  , reinforcing the idea that specific nonlinear effects may be central to their functional dynamics  . as a matter of fact ,anharmonic effects are known since a long time to be highly relevant in many dynamical processes in proteins  .for example , recent experiments show that the activation of anharmonic modes is required for enzymatic activity in lysozyme  .+ more recently , it has been shown that nonlinear excitations known in many systems as discrete breathers ( db ) allow one to cleverly dissect hot - spot sites and intramolecular signaling pathways of connected residues critical to protein functioning  .dbs are time - periodic , spatially localized vibrational modes that are found generically in many - body nonlinear systems  and possess many properties that make them interesting in the context of protein dynamics and allosteric communication .it has been shown that dbs promote and sustain long - range dynamical cross - talk in proteins , mediating energy transfer to distant locations  .dbs are able to self - stabilize by harvesting energy from the background , which has also been directly related to db - mediated long - range communication across protein structures  . in fact , at variance with topological excitations such as solitons , interactions between dbs or between a db and a large vibrational energy fluctuation generally cause a substantial flow of energy from the less energetic to the more energetic mode . remarkably , this sheer nonlinear phenomenon has been reported to put distant regions of protein structures in communication  following isolated energy kicks at specific locations that trigger the spontaneous emergence of a db at a distant site  . moreover , by construction db frequencies do not resonate with normal modes , which hampers db - nm resonant transfer and makes dbs robust against perturbations .dbs are also protected from solvent - mediated instabilities , as their hot spots lie at locations typically far from the protein surface  .for all these reasons , dbs appear as ideal candidates to realize specific long - lived , highly - correlated and fold - spanning motions featuring high resilience to perturbations .it should be stressed that , in view of their nonlinear character , dbs , despite being localized in space , are true _ collective _ motions , where all the particles in the system vibrate at one and the same frequency  .this is the deep reason why db - based analyses are able to unveil fold - rooted dynamical pathways of correlated residues over the entire scaffold of a protein  .+ nmr spectroscopy has proved an invaluable tool to dissect long - range correlations in proteins  .recently , methods based on statistical analysis of nmr chemical shift changes  and perturbation patterns  have been introduced , showing a great potential of mapping extended networks of coupled amino acids involved in intramolecular signaling pathways  . in particular , a recent nmr study reported residual dipolar couplings measurements revealing a rather puzzling long - lived correlated motion spanning four strands separated by up to 1.5 nm in ubiquitin  , showing clear signatures on the msec time scale and raising intriguing questions as to the nature of the associated vibrational mode(s ) .the observed correlations resulted from large - scale concerted conformational rearrangements partly mediated by the hydrogen - bonding network .the structures of 640 conformers have been deposited in the pdb repository following this study ( pdb identifier 2kox ) , providing a wealth of precious structural information .+ in this paper we show that the vibrational pattern of several distinct discrete breather modes found in ubiquitin at specific key residues match to an amazing extent the correlated pattern uncovered experimentally .furthermore , we show that the energy required for exciting such modes could in principle be available through thermal energy fluctuations occurring at key sites , typically residing in the stiffest locations of the protein fold .one the one hand , our results strongly suggest that the correlations observed experimentally in ref . could flag the spontaneous excitation of discrete breathers in ubiquitin .more generally , our calculations suggest that db - based methods may be central to unveil sub - structures comprising residues that mediate the transmission of allosteric signals in proteins .+ the paper is organized as follows .first we introduce our coarse - grained model of protein dynamics and describe the essentials of our algorithm for computing approximate analytic db solutions .we then illustrate , in the case of a randomly selected conformer , how dbs localized at specific sites match the experimental correlated pattern .finally , we report the results of the analysis performed over the whole ensemble of nmr conformers , which confirms that the same results apply to dbs computed in all experimentally resolved structures .we introduced the nonlinear network model ( nnm ) with the aim of exploring the subtle effects arising in many - body systems from the interplay of anharmonicity and the lack of translational order ( _ i.e. _ the peculiar 3 protein folds )  , for which little is known  as opposed to nonlinear systems with lattice - like translational invariance  . in the nnma given protein is coarse - grained to the level of amino - acids and modeled as a nonlinear network of identical point - like particles of mass a.m.u .( _ i.e. _ the average amino - acid mass ) placed at the corresponding sites. the nnm potential energy reads \ ] ] where is the distance between particles and and is the separation in the equilibrium structure ( in this case one of the nmr conformers ) . here and denote the instantaneous and equilibrium position vectors of particle , respectively .the connectivity matrix is simply , where is a cutoff that identifies the interacting pairs .the aim being to explore the connection between nonlinearity and the peculiar structural features of the native folds , only c atoms are taken into account and is assumed to be the same for all interacting pairs . following our previous studies  , we take 10 , and fix so that the low - frequency part of the linear spectrum match the corresponding measured frequencies .this gives kcal / mol / .furthermore , we fix kcal / mol/ , which corresponds to a rather weak nonlinearity  the nonlinear - to - linear energy ratio is . from a physical point of view, the constants and can be rationalized in terms of an average inter - residue potential of mean force ( pmf ) . for a given pair of amino acids , a pmf for a given reaction coordinate ( in our case the c-c distance ) can be computed , _e.g. _ via boltzmann inversion from all - atom molecular dynamics simulations .the idea here is that and measure the coefficients of the second - order and fourth - order terms in a taylor expansion of such pmf .it is interesting to note that reaction coordinates associated with side - chain side - chain relative positions ( _ e.g. _ the distance between centers of mass ) are likely to yield strongly anharmonic ( flat ) pmfs .this is a consequence of the strongly nonlinear ( multimodal ) character of side - chain motions as revealed by the dynamics of dihedral angles  .+ in ref . we have introduced a theoretical protocol for calculating analytically db modes of given amplitude at a given site in the framework of the nnm . in principle , like all time - periodic functions , a db solution can be decomposed as a fourier series comprising harmonics of a fundamental frequency. however , the idea is to start from an _ ansatz _ for the db consisting of the simplest time - periodic function modulated by a time - varying amplitude where we assume that the spatially localized envelope function varies slowly on the timescale defined by the inverse db frequency .this fact can be simply pictured as an alternative statement of db resilience to perturbations  .moreover , we also assume that , so that the parameter sets the physical scale for the oscillation amplitude . under these hypotheses, we can substitute ansatz  in the equations of motion and expand the forces in power series of {a=0 } \ ! \ ! \!a^{p } + \mathcal{o}(\epsilon^{4})\ ] ] we then multiply eqs .  by and average over one db period . under the slowly - varying envelope approximation , we can thus eliminate the time dependence , _ i.e. _ where denotes the time average . neglecting by the same token the second time derivatives of the functions , we finally map the original set of differential equations onto a nonlinear system of algebraic equations , whose unknowns are the time - averaged envelope patterns ( normalized to the displacement of the central particle ) and the breather frequency . a db mode with given amplitude is then found by numerically solving the algebraic system corresponding to an appropriate initial guess .we have found that the local direction of maximum stiffness provides an excellent initial guess , ensuring fast convergence and allowing one to investigate the parameter regions where a db mode localized at a given site can be found  .a general structural analysis proves useful as a start , as this will provide an instructive reading frame for the results illustrated in the rest of the paper .the pdb file labelled 2kox contains 640 possible conformers of ubiquitin .thus , in principle a nnm can be constructed starting from each of those structures and analyzed separately .we have shown that few dbs localized at specific key sites , usually within stiff locations , appear to be far more stable and robust than all other dbs .such dbs arise as nonlinear continuations of high - frequency normal modes , which are spatially localized too , and invariably feature gap - less excitation spectra at variance with dbs localized at generic sites  .this means that we can gain valuable insight by examining the patterns of hot - spot sites highlighted by high - frequency nms within the structural network of the protein over the whole nmr ensemble .+   ( color online ) probability that a residue be the nm site ( _ i.e. _ the particle with the largest amplitude ) of the highest- or second - highest - frequency normal mode .the cartoon shows a c-coarse - grained structures of ubiquitin .the region spanning the strands where the correlated motion is detected in the nmr experiments is highlighted in brown .residues leu 50 and leu 69 are shown as yellow and dark blue spheres , respectively .the cutoff used to calculate the connectivity matrices is .,width=11 ]  for each conformer , we construct a network as indicated above and compute the normal modes , _i.e. _ the eigenvectors of the hessian matrix of the total potential energy  . fig .[ f : conn - nm12 ] reports the pattern of occurrence of the _ nm sites _ of the first two high - frequency nms , _i.e. _ the particles whose amplitude of vibration in the mode is largest .it is apparent that a few sites stand out over the whole ensemble , which identify rigid , hinge - like locations  .these sites lie within the stiffest regions and are thus subject to small - amplitude fluctuations within the ensemble of conformers .overall , hot - spot sites are identified around lys 27 , leu 50 and leu 69 .interestingly , according to nmr measurements reported in ref . , the last two sites appear to participate to the correlated motion uncovered in the experiment .more precisely , the latter identify the following subset of residues ( see also cartoon in fig .[ f : conn - nm12 ] ) our analysis over the ensemble of conformers clearly singles out three different regions as possible localization hot - spots for nonlinear localized modes .thus , the question naturally arises as to what is the correlated pattern associated with dbs localized at these special locations .+ figures [ f:0400 ] ( a ) and ( b ) show the theoretical displacement patterns of two db modes centered at leu 67 and leu 69 .these have been calculated within the nnm starting from the equilibrium structure corresponding to a randomly picked conformer within the nmr ensemble ( equivalent pattern maps are obtained by selecting other conformers ) .it can be clearly appreciated that such db modes could be regarded as plausible realizations of the experimentally detected motions , as the displacement patterns match to a remarkable extent the experimental observations .+   ( color online ) frequency vs energy for different discrete breathers computed in the 400-th conformer .the horizontal dashed lines mark the frequencies of the first edge normal modes.,width=10 ]  as a general fact , the frequency and energy ranges where dbs can be found in a protein are broad and strongly site - dependent  .the dispersion curves shown in fig [ f : we400 ] make clear that pattern - matching dbs such as those illustrated in fig .[ f:0400 ] can be excited at energies as low as 5 - 10 kcal / mol at selected sites with frequency above the linear spectrum .this places such modes among the most robust ones , as resonances with nms are only possible with higher harmonics of linear modes . to be more precise , pattern - matching dbs such as those centered at phe 45 and leu 69 happen to fall within the special class of _ zero - gap _ modes , that is , they can be excited at arbitrary low energies , and can by all means be regarded as analytical continuations of band - edge normal modes that are stabilized through non - linear mechanisms  and 3 nms ( from the band edge ) , respectively . ] .remarkably , this also means that they are among the most stable dbs the more energy is injected the more localized and resilient they become  . at low energies ,such modes are in fact _ intra - band _ breathers , _e.g. _ their frequencies fall within the gap between two successive normal modes .this is made possible by the _discrete _ nature ( finite number of particles ) of the protein and constitutes one of the most distinctive features of localized nonlinear modes in discrete systems lacking translational symmetry  .+ as a general fact concerning breathers in protein structures  , the majority of sites does not host zero - gap dbs , meaning that it can be exceedingly hard to excite a db at a generic location , depending on the associated energy threshold . on the contrary, zero - gap dbs can only be found at very few special sites , usually lying within the stiffest regions of the structure  .it is thus tempting to speculate that a particular biological relevance can be attached to the patterns of selected zero - gap dbs , which are likely to be excited spontaneously at 300 k . following ref . , the average waiting time between spontaneously occurring uncorrelated energy fluctuations of magnitude can be estimated as where and ps is a typical damping coefficient specifying the decorrelation time scale of atomic tumbling . for fluctuations in the range / mol at k , one gets nsec .this means that it should be possible to observe the signature of a db excited out of thermal fluctuations ( at least ) at sites leu 69 and phe 45 .therefore , the possibility that db excitation may explain the correlated motions observed experimentally in ref . appears physically realistic .we have seen that for a randomly selected conformer the vibrational patterns of db modes centered at specific sites match to a surprising extent the correlated motion found experimentally . however , our calculations take a single conformer as the equilibrium structure for constructing the nnm .therefore , it is necessary to investigate whether this is an isolated property displayed by db modes in a few special conformers , or rather it reflects a general property of db solutions over the ensemble of ( experimentally determined ) allowed conformations .we note that part of the answer is already known , as the few key sites where the interesting dbs appear lie within stiff regions .therefore , they should retain similar spatial arrangements of their neighborhoods over the ensemble ( see again fig . [f : conn - nm12 ] ) . this should guarantee that similar dbs should be found at the same hinge - like locations in all the conformers .+ to answer the above question , we introduce the following energy - dependent indicator for a specific db whose vibrational pattern is  where is the subset of residues that participate to the long - lived experimental motion specified in ( [ e : subseta ] ) . by construction ,the quantity measures to what extent the pattern of a given db ( with a given energy ) in a given conformer matches the experimentally highlighted structural correlation .it should be noted that only measures the _ geographical _ overlap , ignoring by construction the specific directions of the vibrational patterns to be compared , as this information is unfortunately not available from the experiments .+ of course , it could be objected that there exists a given probability that _ whatever _ pattern match a given region comprising five non - overlapping segments of the protein scaffold . for this reason, we have also calculated a associated with such null hypothesis , _i.e. _ a measure of statistical significance associated with the measured values .to this aim , the following quantity has been also computed at all values of energies for all conformers where the probability distribution has been reconstructed by generating a large number of random partitions , each consisting of a random permutation of five segments of fixed length ( the same lengths as the segments in ) centered at as many random locations .the quantity is simply the probability that a given match score larger than the one actually observed would be observed for a random partition with the same structure as .thus , provides an estimate of the rejection probability associated with the corresponding value .the results of these analyses are shown in figs .[ f : faall50 - 69 ] .+ as a first global observation , it can be appreciated that the fraction of sizable pattern - matching scores is large over the whole ensemble of conformers .furthermore , while at low energies dbs in different conformers display substantially variable scores , above 10 kcal / mol all dbs display the same large value of . as the pattern - matching score of dbs increases, it can be seen that this is so with increasing statistical confidence , as the corresponding rejection probability drops to zero .the case of the db centered at leu 69 provides a rather clear demonstration of this effect .+ it is interesting to remark that in some cases the fingerprint of the experimental correlated pattern seems to be present _ in nuce _ already at the harmonic level .this can be clearly appreciated by comparing the histograms of values for dbs with energies between 0.1 and 1 kcal / mol and between 7 and 10 kcal / mol . in the case of dbs localized at leu 69 , for example ( fig .[ f : faall50 - 69 ] ( a ) ) , nonlinearity manifestly causes a pre - existing , low - energy signature to become sharper .as the db becomes more energetic , its pattern captures to an increasing extent the experimental correlated motion . in this case, one may recognize nonlinear focussing of a pre - existent , fold - encoded vibrational pattern .+ however , as it is seen from fig .[ f : faall50 - 69 ] ( b ) , nonlinearity is also able to promote a localized mode capturing the experimental pattern at intermediate and high energies without it showing significant traces in the harmonic regime . at variance with dbs localized at leu 69 , dbs centered at leu 50 change substantially from low to intermediate energies , increasingly focussing their vibrational pattern within the experimental region . in this case, a _ dynamical rearrangement _ occurs through sheer nonlinear effects , as a db changes markedly its pattern to match the experimental vibration to an increasing extent when its energy builds up .this result demonstrates in a clear fashion that our nonlinear analysis is able to dig up information unavailable at the nm level .+ altogether , the physical properties of the same dbs in different conformers show to be highly consistent .this is demonstrated by the energy vs amplitude relations displayed in fig .[ f : we ] , which makes clear that the mean curves calculated by averaging the corresponding single - conformer relations over the ensemble are in excellent agreement with the properties of the _ average db _ , _i.e. _ the db computed in the ensemble - averaged structure .furthermore , we observe that the same gap - less dbs are present in a sizable fraction of the conformers ( as _ e.g. _ in the case of the db at leu 69 whose pattern is shown in fig .[ f:0400 ] ) .considering for example the dbs at leu 69 and leu 50 , we find that 39 % and 46 % of the dbs over the ensemble , respectively , display a vanishing gap .the average excitation threshold for the rest of the breathers is ( leu 69 ) kcal / mol and ( leu 50 ) kcal / mol ( see also insets in fig . [f : we ] ) .all in all , it seems that it is easier to excite dbs whose pattern bears less resemblance to those of high - frequency normal modes .in this paper we show that the puzzling features of a long - lived , highly correlated motion found in ubiquitin through nmr measurements  match to a surprising extent the vibrational pattern of discrete breather modes centered at specific hot - spot sites .our calculations show that such nonlinear modes could be excited spontaneously out of thermal fluctuations at room temperature .while this could be the first demonstration of the experimental detection of db - like vibrations in a protein , it is at the same time a powerful demonstration of the ability of our nonlinear analysis  to predict relevant structure - spanning dynamical structures that may be central to the transmission of allosteric signals across protein scaffolds  .+ the emerging picture is that persistent concentration of energy along specific patterns could play a pivotal role in directing conformational changes at a higher level . one may speculate that protein folds might encode for the repeated excitation of long - lived bursts of higher - than thermal energy localized at specific hot spots located within stiff regions .therefore , pattern - matching large - scale correlations embodied by collective modes could be effectively sustained by such long - lived nonlinear vibrations , that tend to self - focus where the nodes of the extended modes are .we note that this picture is intriguingly remindful of the interplay between nonlinear localized and delocalized motions found by garcia in atomistic molecular dynamics trajectories of a small protein in the early 1990s  .+ in this scenario , already evoked in the context of enzyme functioning  and allosteric behavior  , the node / hot spot pattern complementarity of specific slow and fast modes would lie at the core of protein functional dynamics .specific , fold - encoded large - scale motions could be stabilized through nonlinear effects , thus causing the protein to privilege specific functionally relevant fluctuations , such as those of a binding pocket , hinged at some hot - spot site(s ) , that intermittently but steadily breaths open or analogous large - scale motions involving cross - talk between hinged domains .this picture is ultimately supported by general properties of discrete breathers : dbs are not only known for their resilience to perturbations , low - damping rate due to reduced contact with the solvent and hindered resonant energy transfer with nms .remarkably they display the unique property of self - sustaining by _ harvesting _ energy from the background  , which has been recently demonstrated to also occur in proteins  .+  in summary , our study prompts the intriguing hypothesis that unusually long - lived db - like modes might be central to rationalize how protein folds encode intramolecular cross - talk . as such , db - based analyses could provide a key computational method to identify unknown dynamical structures at the core of allosteric transduction mechanisms in proteins .the author is deeply thankful to r. nussinov , p. csermely , y .- h . sanejouand and p. de los rios for a critical reading of this manuscript and for their most enlightening comments . the author is also indebted to r. b. fenwick , l. orellana and x. salvatella for illuminating discussions concerning the application of the db - based method to their nmr data .finally , the author would like to thank l. turin for making him aware of the inspiring ideas developed in the early 1970s by c. w. f. mcclare .the author acknowledges financial support from the eu - fp7 project papets ( ga 323901 ) .

Abstract: in this paper we propose a novel theoretical framework for interpreting long - range dynamical correlations unveiled in proteins through nmr measurements . the theoretical rationale relies on the hypothesis that correlated motions in proteins may be reconstructed as large - scale , collective modes sustained by long - lived nonlinear vibrations known as discrete breathers ( db ) localized at key , hot - spot sites . dbs are spatially localized modes , whose nonlinear nature hinders resonant coupling with the normal modes , thus conferring them long lifetimes as compared to normal modes . dbs have been predicted to exist in proteins , localized at few _ hot - spot _ residues typically within the stiffest portions of the structure . we compute db modes analytically in the framework of the nonlinear network model , showing that the displacement patterns of many dbs localized at key sites match to a remarkable extent the experimentally uncovered correlation blueprint . the computed dispersion relations prove that it is physically possible for some of these dbs to be excited out of thermal fluctuations at room temperature . + based on our calculations , we speculate that transient energy redistribution among the vibrational modes in a protein might favor the emergence of db - like bursts of long - lived energy at hot - spot sites with lifetimes in the ns range , able to sustain critical , function - encoding correlated motions . more generally , our calculations provide a novel quantitative tool to predict fold - spanning dynamical pathways of correlated residues that may be central to allosteric cross - talk in proteins .


at present , one of main problems facing scientists in the field of artificial intelligence ( ai ) is the development of intelligent information systems ( iis ) to solve problems arising in different ai fields .the solution of such a problem is often reduced to heuristic programming that can yield good results , but , in this case , the complex solution of the problem is not provided .systems based on knowledge representation models ( krms ) belong to largest classes of iiss . to date , the following models are well - known and actively used : semantic networks , conceptual graphs , frames , scripts , logical and production models , ontologies , etc .but , despite the use of some krm or other as the base for an iis , an implemented system consists of at least two levels , namely , the krm level and level of its practical implementation .an implementation of a krm often creates definite problems and complexities connected with interacting different levels of the system . in this connection ,an object - oriented knowledge representation model was proposed in , which can be integrated into object - oriented programming languages and thereby to unite the krm itself and the language used for its implementation .this approach will allow one to get rid of some levels of abstraction and to partly simplify the structure of the system being developed and , hence , the development process itself .the present article describes main ideas that form the basis for constructing fuzzy object - oriented dynamic networks , in particular , fuzzy objects and classes that allow one to structurally represent knowledge about objects that are unfuzzily specified , are fuzzy , or are incompletely defined and , at the same time , to classify them .operations over fuzzy objects and classes of fuzzy objects are also considered with the help of which sets and new classes of fuzzy objects can be constructed and , thereby , new knowledge can be obtained .objects can be considered to be both arbitrary things from our real life and results of using our imagination . in other words ,an object is every thing that can be perceived in some way or other , i.e. , can be sensed or imagined .it is obvious that each object ( irrespective of its nature ) has definite properties that are characteristic for it .for example , let us consider a natural number .it is obvious that it must be integer and larger than zero .it is these characteristic properties of natural numbers that allow one to distinguish them from other objects .in fact , in order to reveal whether the numbers , , and are natural , it is necessary to check their properties , in particular , to check whether these numbers possess the same properties as natural numbers . as a result of verification ,it becomes obvious that and are not natural numbers . from thisit is possible to draw the conclusion that a natural number is a crisply defined object .but , in addition to crisp objects , there also are other objects such as fuzzy and approximately or incompletely described objects .they arise in trying to recollect something , in describing our desires or fancies , in searching for something about which we know very little , etc .they mentally arise when we want to formalize some intuitive guess whose whole nature is fuzzy from the viewpoint of logic or mathematics .l. a. zadeh was the first to propose a formalistic approach to the description of such objects . with time , this approach was transformed into a theory , and many results were obtained within the framework of this theory , in particular , in the field of constructing information systems that operate with fuzzy concepts . taking into account the flexibility and efficiency of the approach proposed by l. zadeh , we use here some of his ideas for formal definition of fuzzy objects .in addition to the aforesaid concerning objects and their properties , we will pay attention to the following important point : properties of an object and the object itself are closely interconnected and can not exist separately .properties do not exist in themselves without an object since some object is their manifestation , and properties can not be seen , understood , and even described without it . in turn , an object can not exist without properties since their absence makes the construction or even a representation of the object impossible .it is obvious that this takes place for both crisp and fuzzy objects .therefore , a formal definition of the concept of an object can not be formulated if its properties are not formally defined and vice versa , it is impossible to define properties without knowing an object . in this connection ,we first define properties of an object and then the object itself with allowance made for their interrelations .  before considering properties of an object ,note that they are subdivided into quantitative and qualitative ones .a fuzzy quantitative property of an object is a tuple of the form , where is a fuzzy set describing the quantitative value of the property and are its measurement units .* example 1 * let us consider an object , for example , an apple ; one of its properties is its mass .after weighing it , we obtain its exact mass , and this will determine the value of its crisp quantitative property described in .but if it is impossible to approximately or exactly weigh an apple and thereby determine its mass , then we can represent it using a fuzzy set .if we take an apple and , based on our sensation , reveal that its mass is about , then this mass can be represented in the form   let us define the equivalence of two fuzzy quantitative properties of objects to provide the possibility of comparing them .two fuzzy quantitative properties and are equivalent , i.e. , , if and only if the following conditions are fulfilled :  1 .  ; 2 . , , ; 3 . , , .a fuzzy qualitative property of an object is a verification function  ] .let us define the equivalence of two fuzzy qualitative properties of objects to provide the possibility of comparing them .two fuzzy qualitative properties and are equivalent , i.e. , , if and only if the following condition is fulfilled :   since one object can have several properties that are quantitative and qualitative , it makes sense to define the concept of a fuzzy object specification .  a specification of a fuzzy object is a vector of the form , where , , is a quantitative or qualitative property of the object .  in other words , the specification of a fuzzy object can consist of crisply specified and fuzzy properties. moreover , qualitative properties are special cases of fuzzy qualitative properties .next , using the concept of a fuzzy object specification , the concept itself of a fuzzy object can be directly defined .a fuzzy object is a pair of the form , where is the object identifier and is its specification .* example 3 * we consider an object , namely , a water - melon that has , as is well known , a spherical form and assume that its weight is about . using the definition of a fuzzy object , it is possible to formally represent this water - melon as , where , , and   it is obvious that the specification of the object w can have a larger number of properties , which depends on the level of detail to be taken into account in considering the object .in addition to object properties , methods ( operations ) that can be applied to them should be taken into account , which makes it possible to operate with these objects to a certain degree . from thisit makes sense to define the concept of a method of an object .an operation ( method ) of a fuzzy object is a function that can be applied to the object with allowance for distinctive features of its specification .depending on the nature of an action on an object , methods can be divided into the following two types : modifiers , i.e. , functions that can change the object , in particular , values of its properties , and exploiters , i.e. , functions using objects as unchangeable parameters .  * example 4 * consider a fuzzy object , namely , a square that is introduced by its specification , where is the number of its sides , is the number of its angles , is the size of its sides , and is the degree measure of its angles .an example of an exploiter for the square is the function for computing its area , i.e. , , where . as a result, we obtain in this case , an example of a modifier can be the following function of increasing a square whose application promotes the increase in the size of its sides : , where is some natural number . as a result , we obtain since several methods can be applied to the same object , it makes sense to define the concept of the signature of a fuzzy object .a signature of a fuzzy object is a vector of the form , where , , is a method of the object .next , for purposes of comparison , we will define the equivalence of two fuzzy objects .two fuzzy objects and are considered to be of the same type if and only if they have equivalent specifications and the same methods can be applied to them , i.e. , and .  it is obvious that each object , regardless of its nature , belongs to at least one class . in this connection , we will define the concept of a class of fuzzy objects .a class of fuzzy objects is a tuple of the form , where is a specification when several fuzzy objects are described and is their signature .  by a class of fuzzy objects we understand properties of objects and methods that can be applied to them . in other words ,a class of fuzzy objects is a generalized form of consideration of a number of fuzzy objects without the objects themselves . analyzing the definition of a class of fuzzy objects, one can draw the following conclusion : in creating objects of this class , all fuzzy quantitative properties can be represented in the form of fuzzy sets of the second type .  as well as in the case of classes of crisply specified objects , classes of fuzzy objectscan be divided into two types , namely , homogeneous and heterogeneous .reasons for this partition , distinctive features , and examples of these types of classes of objects are described in detail in , .therefore , we pass directly to the definitions of the concepts of homogeneous and heterogeneous classes of fuzzy objects .a homogeneous class of fuzzy objects is a fuzzy class of objects that describes only fuzzy objectsthe same type .* example 5 * any classes of convex polygons such as those of squares , rectangles , triangles , etc .are homogeneous classes of fuzzy objects .a heterogeneous class of fuzzy objects is a tuple of the form , where is the core of the class of objects that consists of only the properties and methods that are common to specifications and signatures , respectively ; , , are the projections of objects consisting of only the properties and methods that are peculiar to only fuzzy objects .* example 6 * classes of all convex polygons , classes of all cars with the same brand name , the class of all tvs of one producer , etc .are heterogeneous classes of fuzzy objects .a distinctive feature of all the methods mentioned above , irrespective of the class in which they are defined , is that all of them are local , i.e. , are closed under the class in which they are defined .the reason is that methods of objects are intrinsically defined with allowance for specifications of objects , i.e. , proceeding from properties .there are methods that are defined in one class of fuzzy objects but , at the same time , can be applied to other classes of fuzzy objects , i.e. , they are polymorphous within certain limits . for implementing polymorphism, some object - oriented programming languages use the mechanism called operator overloading .  in ,operations such as union , intersection , difference , symmetric difference , and cloning are considered that can be applied to any objects and classes of objects ; at the same time , they do not require overloading and are rather universal .important distinctive features of these methods are their set - theoretic ( object - based ) character and the possibility of obtaining new objects and classes of objects with their help .this is directly related to the important task rcg ( runtime class generation ) , i.e. , the generation of classes during program execution .despite the fact that all these operations were defined for crisply specified objects and classes of objects , they also can be applied to fuzzy objects and classes of fuzzy objects . here, we will not define the operations themselves since they are similar for the case of fuzzy objects and classes of fuzzy objects .let us consider an example of their practical application .* example 7 * consider geometric figures , for example , a square and a rhombus .it is obvious that they are representatives of different classes of convex polygons .we define their classes as follows : where and are the numbers of sides and and are the sizes of sides ,  and are the numbers of angles of the figures ; and are the grade measures of angles ; and signify the equality of all sides ; and signify the equality of all angles ; and are methods for computing a perimeter ; and are methods for computing an area .  analyzing the classes and , one can state that they are classes of fuzzy squares and rhombuses , respectively , since their specifications contain fuzzy quantitative and qualitative properties.  then we will define the following specifications and signatures of the fuzzy objects and b using the specifications and signatures of their classes :   using the fuzzy square and fuzzy rhombus as arguments , we will consider the operations of union , intersection , difference , symmetric difference , and cloning .  _ union operation _ .  as a result ,a set of fuzzy objects and a new class of fuzzy objects are obtained .here , where is the number of sides ; is the number of angles , signifies the equality of all sides , and signifies a method for computing perimeters ; this allows us to draw the conclusion that is the set of fuzzy squares of type and fuzzy rhombuses of type and that the class is a heterogeneous class of fuzzy objects and simultaneously describes two types of figures , namely , and ._ intersection operation _ .  as a result ,a new class of fuzzy objects is obtained , where all the properties contained in coincide with the properties from the core of the class obtained as a result of union of the fuzzy objects and . an analysis of the result does not allow one to exactly determine the type of geometric figures that describes the class of fuzzy objects but , in this case , it may be said that it is homogeneous and consists of properties that are common to fuzzy squares of the class and fuzzy rhombuses of the class ._ difference operation _ .  as a result, we obtain a new class of fuzzy objects all properties of this class are exactly repeated in the projection of the object that is obtained as a result of union of the fuzzy objects and , i.e. , it is obvious from the analysis of the result that , in contrast to the previous case , the homogeneous class of fuzzy objects describes a fuzzy rhombus and uses a smaller specification in this case ._ operation of symmetric difference _ .  as a result, we obtain a new class of fuzzy objects , where the projections are similar to the projections obtained as a result of union of fuzzy objects and . analyzing the result, one can draw the conclusion that the heterogeneous class of fuzzy objects describes two types of geometric figures one of which can be a fuzzy rhombus and the second can be a fuzzy square or a rectangle .  _ cloning operation _ .  as a result , a new numbered copy of the fuzzy object is obtained .it is obvious from example 7 that all the considered operations over fuzzy objects are exploiters since they do not change the fuzzy objects and but only use them as parameters . in this connection, we pass to the consideration of another type of operations over fuzzy objects , namely , modifiers .a detailed definition of a modifier of objects is given in , where five types of modifiers ( complete , partial , generating , destroying , and replacing ) are considered and the principle of construction of combined modifiers on their basis is shown .in , the structures of objects and classes of objects are analyzed and also structural interrelations between properties of objects and classes of objects within the framework of their specifications are shown .these interrelations play an important role since they can be violated in the case of a modification and , as a result , the model does not necessarily correspond to real objects whose changes should be modeled . in , this phenomenon is called the principle of reflection according to which changes in some property are rather frequently impossible in nature without a definite action on other properties that are connected with the former one .example 7 partially demonstrates this principle since the properties and directly depend on values of the properties and , respectively .it follows from this that a modification of the properties and sometimes stipulates a change in the properties and since , otherwise , this model will cease to describe the process being considered ; moreover , the model becomes inconsistent .assume that we modify the property by specifying different lengths of figure sides .if we do not modify the property in this case , then inconsistency will be obtained .processes of modification of crisply specified and fuzzy objects or classes of objects are underlain by the same principle , and a distinction lies only in the crispness or fuzziness of a specification . in this connection, we will give an example of modifications of a fuzzy object .* example 8 * we consider the fuzzy square from the previous example and modify it so that , as a result , a fuzzy rhombus is obtained . to this end, we will construct a partial modifier . here , , and are functions of modification of the properties , , and , respectively .thus , the fuzzy object is transformed into a fuzzy rhombus under the action of the modifier .it should be noted that this modification of the object also leads to a change in the signature of its class since the method is incorrect for the fuzzy object and , therefore , the method becomes undefined . analyzing the result of this modification , one can draw the conclusion that a modification of objects and classes of objects stipulates the creation of new classes of objects , which also is directly related to rcg .this example demonstrates only some of many aspects of the process of modification of objects and classes of objects .note that all well - known operations over fuzzy sets can be used as functions for modifying fuzzy properties of objects .let us consider the corresponding example .* example 9 * as an object , we use the fuzzy rhombus from example 7 . consider the following partial modifiers : where are functions for modifying the property and and are natural numbers . in this case , the modifiers and are operations of dilution and concentration of fuzzy sets  .this article formulates definitions of a fuzzy object and a class of fuzzy objects that can underlie structural descriptions of fuzzy objects and , at the same time , allow for their classification .two types of operations over fuzzy objects and classes of fuzzy objects ( exploiters and modifiers ) are also considered with the help of which sets and new classes of objects can be created and also changes in structures of objects and their properties can be modeled , in particular , under the influence of external factors .this approach allows one to model some capabilities of human intelligence , in particular , mechanisms of analysis , classification , search , and recognition of objects and classes of objects on the basis of their features .the results obtained allow us to testify that fuzzy object - oriented dynamic networks are constructed as models for the representation of knowledge on fuzzy objects , classes , and concepts .terletskyi , d.o . ,provotar , o.i . : object - oriented dynamic networks . in : setlak , g. , markov , k. ( eds . ) computational models for business and engineering domains .ibs isc , vol .30 , pp . 123136 .ithea ( 2014 )   terletskyi , d.a . ,provotar , o.i . : mathematical foundations for designing and development of intelligent systems of information analysis .scientific journal `` problems in programming '' .15 , 233241 ( 2014 )    terletskyi , d.o ., provotar , o.i . :class and object modifiers : in proc .xiith all - ukrainian .sci .- pract . conf . of students , graduate students , and young scientists `` theoretical and applied problems of physics , mathematics , and informatics '' [ in ukrainian ] , vpivpk `` politekhnika '' , kyiv , 8587 ( 2014 )

Abstract: the concepts of fuzzy objects and their classes are described that make it possible to structurally represent knowledge about fuzzy and partially - defined objects and their classes . operations over such objects and classes are also proposed that make it possible to obtain sets and new classes of fuzzy objects and also to model variations in object structures under the influence of external factors .


consider a game in which the player s goal is to force a brownian particle out of an interval ( say  ] defined by , with associated expectations .here , is the drift that applies just before time , and which can change at time precisely , if desirable .let be the _ switching cost _ and let : a_{s-}\neq a_{s}\}\ ] ] be the number of switches of drift of the process up to time .notice that is possible .the _ cost function _ for the minimization ( resp .maximization ) problem is then given by respectively , where ,1[\}\ ] ] and the _ value functions _ are respectively given by the goal is then to compute these value functions and to find _ optimal controls _ and such that and , for all and for all .   in order to formulate a free boundary problem for each value function, we will assume that the solution will satisfy three properties .the validity of these properties will be established in section [ sec : proofs ] . +* property 1 . * _ the optimal strategy is symmetric with respect to the initial drift and the value functions satisfy _ _ and _ , _ for all _ ] .thus , if the cost exceeds a certain value , then a reasonable player will never pay this cost to change the initial drift .this value is given by the maximal difference between the expected exit time from  ] and .then , by taking the derivative at of the moment - generating function of a brownian motion with drift ( see ( * ? ? ?* ii.2.3 ) ) , we find , after tedious calculations , that and that }\left(f^\mu(x)-f^{-\mu}(x)\right)=\max_{x\in[0,1]}\left ( -\frac{2x}{\mu}+\frac{1}{\mu}+\frac{e^{2\mu x}-e^{2\mu(1-x)}}{\mu(e^{2\mu}-1)}\right)\nonumber\\ & = \frac{-1}{\mu^2}\left\{\log\left(\tfrac{\sinh(\mu)}{\mu}\left(1-\sqrt{1-\tfrac{\mu^2}{\sinh^2(\mu)}}\right ) \right)+\sqrt{1-\tfrac{\mu^2}{\sinh^2(\mu)}}\right\}\,,\label{eq : def : c*}\end{aligned}\ ] ] for any ( note that and ) .this maximum is attained at   the third property concerns the general shape of the optimal strategy .indeed , consider two scenarios in the minimization problem .assume first that one starts near with a positive drift .the player will keep this favorable drift for a while . if the particle goes down , then when it reaches , both drifts are equivalent because of the symmetry property . since the player is subject to a switching penalty, he will keep the positive drift .if the particle keeps going down , then it will become more advantageous to change to a negative drift so that the particle will exit more quickly through .secondly , if one starts close to with a positive drift , then the diffusive behavior of the particle makes it very likely that it will rapidly hit even if the drift is in the unfavorable direction .thus , it is probably not worthwhile to pay the penalty to change the drift .these two facts are summarized by the following property : + * property . *_ there exist two barriers and satisfying and such that it is optimal to keep a positive drift above or below and it is optimal to switch to a negative drift within  ] . _   via the _ the dynamic programming principle _ and the hamilton - jacobi - bellman equation ( see e.g. ( * ? ? ?* section 3.1 ) , ( * ? ? ?* section 3 ) , or ( * ? ? ? * chapter 5 ) ) , the value function of the stochastic control problem ( [ eq : control : problem : min ] ) should satisfy the following system of variational inequalities : ,1[\,,\\ v_{c}(0,\pm1)=v_{c}(1,\pm 1)&=0.\end{aligned}\ ] ] we formulate this system as a _ free boundary problem _ for the value function . in the region where it is optimal to keep the current drift , the value function must satisfy the ordinary differential equation ( [ eq : ode : continuation ] ) below . in the region where it is optimal to switch to the other drift, we have the equation ( [ eq : switching ] ) . with the three properties of section [ sec : properties ], we expect that the value function should satisfy the following problem , in which ( resp . ) denotes ( resp . ) :  [ eq : fbp : vc ] ,1 ] & & \label{eq : ode : continuation}\\ v_c(x,1)&= v_c(x,-1)+ c , & & x\in[a_c , b_c ] & & \label{eq : switching } \\v_c(0,1)&= v_c(1,1)=0 , & & & & \mbox{(boundary conditions)}\label{eq : boundary : cond } \\v_c(a_c-,1)&= v_c(a_c+,1 ) , & & & & \mbox{(continuous fit ) } \label{eq : fit1 } \\v_c(b_c-,1)&= v_c(b_c+,1 ) , & & & & \mbox{(continuous fit ) } \label{eq : fit2}\\ \frac{\partial v_c}{\partial x}(a_c-,1)&= \frac{\partial v_c}{\partial x}(a_c+,1 ) , & & & & \mbox{(smooth fit)}\label{eq : smooth : fit1}\\ \frac{\partial v_c}{\partial x}(b_c-,1)&= \frac{\partial v_c}{\partial x}(b_c+,1 ) , & & & & \mbox{(smooth fit)}\label{eq : smooth : fit2}\\ v_c(x,-1)&=v_c(1-x,1 ) ,  & & x\in[0,1 ]  & & \mbox{(symmetry)}\label{eq : symmetry}\end{aligned}\ ] ]  where and are two unknowns satisfying .[ prop : sol : fbp : min ] let be given by ( [ eq : def : c * ] ) .there exists a unique solution to the free boundary problem ( [ eq : fbp : vc ] ) .  1 .if , then the solution is given by and ,\,a\in\{\pm1\},\ ] ] where is the constant strategy and is defined by ( [ eq : def : fnu ] ) .moreover , }\left\{f^{\mu}(x)-f^{-\mu}(x)\right\}\quad\text { and } \quad f^\mu(a_{c^*})=f^{-\mu}(a_{c^*})+c^*(\mu).\ ] ] 2 .if , then the solution is given by , \\\displaystyle  \tfrac{1-x}{\mu}+\alpha_c \left(e^{2\mu(1-x)}-1\right ) , & x\in\,]b_c,1 ] , \\[1ex ]       \end{array }      \right.\\      \bar{v}_c(x,-1)&=\,\bar{v}_c(1-x,1),\hspace{18ex } x\in[0,1],\nonumber     \end{aligned}\ ] ] where is the unique solution of the transcendental equation  is the unique solution ,b_c[ ] : ,1 ] ,    & & c_{-1}= [ 0,1-b_c[\,\cup\,]1-a_c,1],\\ d_1&= [ a_c , b_c ] ,           & & d_{-1}= [ 1-b_c,1-a_c].\end{aligned}\ ] ] the subsets and are called respectively the _ continuation _ and the _ switching _ region for the drift . for , we define the candidate optimal strategy as follows . let \times\{-1,1\} ] . then define , for , and for , construction implies that satisfies for all  and the controlled process is the solution of and .observe that the sequence of the stopping times corresponds to the jump times of the strategy , that on and that , the exit time from ,1[ ] and , we have   now that we have exhibited a candidate strategy and a candidate value function as the solution of a free boundary problem , we can state the optimality theorem .[ verification : thm : min ] let be given by ( [ eq : def : c * ] ) .if is the control satisfying ( [ eq : def : candidate : ac : a ] ) ( and ( [ eq : def : candidate : ac ] ) ) and if denotes the unique solution of the free boundary problem ( [ eq : fbp : vc ] ) given in proposition [ prop : sol : fbp : min ] , then :  1 .  for ,the value function is and is an optimal control for the problem ( [ eq : control : problem : min ] ) ; 2 .  for ,the value function is and is -a.s . the unique optimal control for the problem ( [ eq : control : problem : min ] ) .in the case where , we see in proposition [ prop : sol : fbp : min ] that the switching regions consist of one single point , at which a change of drift can be considered as insignificant . indeed at this point , the price to pay for a change of drift is equal to the maximum expected profit provided by this change itself .moreover , the same proposition together with theorem [ verification : thm : min ] show that the constant strategy and the candidate are both optimal in this case .the same remark will apply to the maximization problem . for a uniqueness result when , see proposition [ prop : uniqueness ] .similar to the minimization problem , the value function should satisfy ,1[\,,\\ v^{\text{max}}_{c}(0,\pm1)=v^{\text{max}}_{c}(1,\pm 1)&=0.\end{aligned}\ ] ] in view of the three properties of section [ sec : properties ] , we formulate this system as the following free boundary problem for the value function :  [ eq : fbp : wc ] ^{\text{max}}_c,1 ] \label{eq : continuation : max}\\ & v^{\text{max}}_c(x,1)= v^{\text{max}}_c(x,-1)-c ,            & & x\in[a^{\text{max}}_c , b^{\text{max}}_c ] & &       \label{eq : switching : max } \\ & v^{\text{max}}_c(0,1)= v^{\text{max}}_c(1,1)=0 ,             & & & & \mbox{(boundary conditions)}\label{eq : boundary : condit : max } \\ & v^{\text{max}}_c(a^{\text{max}}_c-,1)= v^{\text{max}}_c(a^{\text{max}}_c+,1 ) ,            & & & & \mbox{(continuous fit ) } \label{eq : cond : fit1:max } \\ & v^{\text{max}}_c(b^{\text{max}}_c-,1)= v^{\text{max}}_c(b^{\text{max}}_c+,1 ) ,            & & & & \mbox{(continuous fit ) } \label{eq : cond : fit2:max}\\ & \frac{\partial v^{\text{max}}_c}{\partial x}(a^{\text{max}}_c-,1)= \frac{\partial v^{\text{max}}_c}{\partial x}(a^{\text{max}}_c+,1 ) , & & & & \mbox{(smooth fit)}\label{eq : smooth : fit1:max}\\ & \frac{\partial v^{\text{max}}_c}{\partial x}(b^{\text{max}}_c-,1)= \frac{\partial v^{\text{max}}_c}{\partial x}(b^{\text{max}}_c+,1 ) , & & & & \mbox{(smooth fit)}\label{eq : smooth : fit2:max}\\ & v^{\text{max}}_c(x,-1)=v^{\text{max}}_c(1-x,1 ) ,       & & x\in[0,1 ]  & & \mbox{(symmetry)}\label{eq : symmetry : max}\end{aligned}\ ] ]  where and are two unknowns satisfying .even though the solution of the free boundary problem is similar to the one in the minimization problem , the two problems are not symmetric as we shall see in the next proposition .[ prop : sol : fbp : max ] let be given by ( [ eq : def : c * ] ) .there exists a unique solution to the free boundary problem ( [ eq : fbp : wc ] ) .  1 .if , then the solution is given by and ,\,a\in\{\pm1\},\ ] ] where and are given by proposition [ prop : sol : fbp : min ] and is defined by ( [ eq : def : fnu ] ) .if , then the solution is given by , \\\displaystyle  \tfrac{1-x}{\mu}+\delta_c \left(e^{2\mu(1-x)}-1\right ) , & x\in\,]b^{\text{max}}_c,1 ] , \\\end{array }      \right.\\      \bar{v}^{\text{max}}_c(x,-1)&=\,\bar{v}^{\text{max}}_c(1-x,1),\hspace{21ex } x\in[0,1],\nonumber     \end{aligned}\ ] ] where and with and given by proposition [ prop : sol : fbp : min ] , is the unique solution ^{\text{max}}_c,1[ ] and , we have   [ verification : thm : max ] let be given by ( [ eq : def : c * ] ) .if is the control satisfying ( [ eq : def : candidate : gc ] ) and if denotes the unique solution of the free boundary problem ( [ eq : fbp : wc ] ) given in proposition [ prop : sol : fbp : max ] , then :  1 .  for ,the value function is and is an optimal control for the problem ( [ eq : control : problem : max ] ) ; 2 .  for ,the value function is , and is -a.s . the unique optimal control for the problem ( [ eq : control : problem : max ] ) .in this section , we solve the boundary problems and and prove that the solution of each is the value function of the associated control problem .we begin with the optimal expulsion problem .the general solution to the o.d.e .( [ eq : ode : continuation ] ) is where and are arbitrary constants . since ( [ eq : ode : continuation ] ) is satisfied on the two disjoint intervals and ,1] ] is obtained by using ( [ eq : switching ] ) together with ( [ eq : symmetry ] ) .we then find that must satisfy , \\\displaystyle  \tfrac{1-x}{\mu}+\alpha_c \left(e^{2\mu(1-x)}-1\right ) , & x\in\,]b_c,1 ] , \\       \end{array }      \right.\ ] ] where and are four unknowns that we have to determine using the equations ( [ eq : fit1])([eq : smooth : fit2 ] ) .they give us after some simplifications , in the same order : multiply ( [ eq : resolutiona ] ) by , ( [ eq : resolutionb ] ) by , ( [ eq : resolutionc ] ) by and ( [ eq : resolutiond ] ) by , to obtain , after simplifications , respectively the four equations subtract ( [ eq : resolutionb : bis ] ) from ( [ eq : resolutiond : bis ] ) , and solve this equation for , then insert this expression into the sum of ( [ eq : resolutionb : bis ] ) and ( [ eq : resolutiond : bis ] ) , to get a new equation ( [ eq : resolution : bc ] ) for . solve ( [ eq : resolutionc : bis ] ) for , and plug this value into ( [ eq : resolutiona : bis ] ) to get a new equation ( [ eq : resolution : ac ] ) for .these equations are and the formulas for and are equations ( [ eq : resolution : bc ] ) for and ( [ eq : resolution : ac ] ) for are transcendental .we define so that and are respectively solutions of and . setting and , we find that solving the system ( [ eq : resolutiona])([eq : resolutiond ] ) with is equivalent to solving with and .[ rem : discussion : h : htilde ] by computing two derivatives , we see that , for all , , therefore , for all , therefore is strictly increasing with and , so that ( [ eq : hct=0 ] ) admits a unique solution -c\mu^2 - 1,-c\mu^2[ ] is obtained by considering the position of the extrema of .indeed , setting and since for all , we have that .thus , the function vanishes three times : the first time on the interval -\infty , 0[\, ] .  it remains to determinethe parameter which , by ( [ eq : beta:2 ] ) , ( [ eq : s0 * ] ) , ( [ eq_def_tc*_2 ] ) and ( [ eq : alpha * ] ) , is given by we have therefore solved the system ( [ eq : hct=0])([eq : beta:2 ] ) and we have found that ,\tfrac{1}{2}\right [ \quad\text { and } \quad a_{c^*}=\frac{s_{c^*}}{2\mu}=\frac{t_{c^*}+\mu}{2\mu}=b_{c^*},\ ] ] with given by ( [ eq_def_tc*_1 ] ) . this establishes the existence and uniqueness of the solution of the free boundary problem ( [ eq : fbp : vc ] ) .this solution is given by , \\\displaystyle  \tfrac{1-x}{\mu}+\alpha_{c^ * } \left(e^{2\mu(1-x)}-1\right ) , & x\in[a_{c^*},1 ] , \\\end{array }      \right .\nonumber\\ & = \left\ {       \begin{array}{ll }       \displaystyle \tfrac{-x}{\mu}+\tfrac{1-e^{-2\mu x}}{\mu(1-e^{-2\mu})}\ , , & \qquad\qquad x\in[0,a_{c^ * } ] , \\\displaystyle  \tfrac{-x}{\mu}+\tfrac{1-e^{-2\mu x}}{\mu(1-e^{-2\mu})}\ , , & \qquad\qquad x\in[a_{c^*},1 ] , \\       \end{array }      \right.\label{eq : value : barvc * } \\ & = f^\mu(x)={\mathbb{e}}_x\left(\sigma^\mu\right ) , \quad\qquad\qquad\ ; x\in[0,1],\nonumber\end{aligned}\ ] ] and . by definition of , we have where is the constant strategy .finally , we find using ( [ eq : ac*=bc * ] ) and ( [ eq_def_tc*_2 ] ) that which , according to ( [ eq : maximum : diff : fnu ] ) , is the location of the global maximum of on the interval  ] and , we conclude that , which establishes ( [ eq : ac ] ) . finally , ( [ eq : beta ] ) is simply a rewriting of ( [ eq : beta:2 ] ) .we now aim to prove that the solution of given in proposition [ prop : sol : fbp : min ] is indeed the value function . for this, we could apply the verification theorem 6.2 of  .however , we prefer to give a direct proof , in which the main steps correspond to checking assumptions of the verification theorem .indeed , lemma [ lem : generator : vbar ] below corresponds to verifying hypotheses ( vi ) and ( x ) of ( * ? ? ?* theorem 6.2 ) , and corollary [ cor : deltah ] below corresponds to hypothesis ( ii ) there .however , by detailing the proof , we can identify where any deviation from the strategy leads to a suboptimal cost function , and this will be useful in our result on uniqueness ( proposition [ prop : uniqueness ] in section [ sec : further ] ) .  before proving proposition [ prop :esperance : tauac ] , we introduce some notations .let be the sequence of switching times of and let be the exit time of from  ] and .notice that for all , on , if is odd and if is even .clearly , and since the process is strong markov by construction , for even ( but also , by symmetry , for odd ) , therefore , that is , given , is a geometric r.v . with parameter ,1[\, ] and are treated similarly .this completes the proof of proposition [ prop : esperance : tauac ] .[ lem : generator : vbar ] let denote the infinitesimal generator of a brownian motion with drift , ; that is , for , { \mathbb{r}}) ] , we have and thus , moreover , since , b_c[ ] , then -b_c,1-a_c[\,\subset\,]b_c,1] ] . as , b_c[\,\subset \,]0,\frac{1}{2}[ ] and , according to ( [ eq : symmetry ] ) and ( [ eq : sol : fbp : min : c < c * ] ) , we will show that for . indeed , vanishes at most twice on , because the equation is a quadratic equation in .the roots of are given by with . by ( [ eq : beta ] ) and ( [ eq : alpha ] ), we have and thus .using , and the fact that , we see that , and so .therefore , . from the formula for and ( [ eq : beta ] ) ,we see that therefore , . by ( [ eq_inequ_2mubc_mtilde ] ) , since , we find that .this implies that the function is monotone decreasing on -\infty , a_c[ ] , then ,1-b_c[ ] .indeed , its derivative vanishes at most twice on , at where . by ( [ eq : alpha ] ), we find that since .hence , , and according to ( [ eq : switching ] ) and condition ( [ eq : smooth : fit2 ] ) , we have that .moreover , we notice that .thus , and the function is monotone increasing on ,1-b_c[ ] , which proves the desired inequality ._ if  ] and by ( [ eq : symmetry ] ) and ( [ eq : switching ] ) , which proves the desired inequality ._ if -a_c,1] ] and since and , the function is strictly positive on this interval .this proves the desired inequality in this last case .let us define \times\{-1,1\}\times{\mathbb{n}}\longrightarrow { \mathbb{r}}_+ ] and , -a.s . , for all : ( 1 ) ; ( 2 ) ; and ( 3 ) if and only if and .let and let . if is a time of continuity of the strategy , then clearly .assume is such that . then and if , then by construction of ( see ( [ eq : switching ] ) ) .if , then by lemma [ lem : ineq : vbar ] .for the control satisfying ( [ eq : def : candidate : ac ] ) , if and only if , and this completes the proof .  we can now prove the following proposition .  [ prop : martingale ] let denote the candidate strategy satisfying ( [ eq : def : candidate : ac ] ) , let be defined by ( [ eq : def : h ] ) and let be given by ( [ eq : def : c * ] ) . if , then for all  ] into , we let ,\ ] ] denote its linear interpolation on \times[-1,1] ] .furthermore , we let be the set of discontinuities of in the variable when .  using the extension of , we see that is in the variables and , in the variable , it is on  ] . applying the _ local time - space _ formula ( * ? ? ?* theorem 3.2 ) , we get for all , where is the local time in of the process . by the smooth fit conditions, we have for all  ] , so the stochastic integral above is a martingale , which establishes the first statement .  on the other hand , for an arbitrary control , we have by the above that for all , where we used lemma [ lem : generator : vbar ] and corollary [ cor : deltah ] .this shows that is a submartingale .we are now ready to prove the optimality of our candidate strategy .let . on one hand , by corollary [ cor : deltah ] and by the first statement of proposition [ prop : martingale] , we have that under , for all . since is a continuous and bounded function , since is an increasing process and since by proposition [ prop : esperance : tauac ] , and , we get by dominated and monotone convergence that as and . on the other hand ,let be such that .then , by corollary [ cor : deltah ] and by the second statement of proposition [ prop : martingale ] , for all . as just above , we get by dominated and monotone convergence that if , it is then clear that . combining ( [ egalite_vbar ] ) and ( [ inegalite_vbar ] ) , we obtain where the infimum is attained by the control .this proves the optimality of the strategy in the case where .. then by definition of the value function . moreover , if denotes the constant strategy , then and , again by definition of the value function , finally , by ( [ eq : sol : fbp : c*:v ] ) and by the first part of the proof , we know that hence , and the strategy is optimal for all .suppose now that and that there exists another optimal strategy such that there exists \times\{-1,1\} ] , there are four constants to determine , which are reduced to two by the boundary conditions ( [ eq : boundary : condit : max ] ) .then , ( [ eq : switching : max ] ) and ( [ eq : symmetry : max ] ) give the form of given in ( [ eq : sol : fbp : max : c < c * ] ) .the four unknowns and have to be determined using ( [ eq : cond : fit1:max])([eq : smooth : fit2:max ] ) . using ( [ eq : sol : fbp : max : c < c * ] ) , these equations give us , after some simplifications , in the same the order : observe that setting and , the equations ( [ eq : resolution : max1 ] ) and ( [ eq : resolution : max3 ] ) become identical to ( [ eq : resolutionb ] ) and ( [ eq : resolutiond ] ) , respectively . in the proof of proposition [ prop :sol : fbp : min ] , we have already established that ( [ eq : resolutionb ] ) and ( [ eq : resolutiond ] ) have a unique solution and .thus , it remains only to prove the existence of and solution of ( [ eq : resolution : max2 ] ) and ( [ eq : resolution : max4 ] ) such that satisfies ( [ eq : delta ] ) and .  equation ( [ eq : delta ] ) follows directly from ( [ eq : resolution : max4 ] ) . substituting ( [ eq : delta ] ) into ( [ eq : resolution : max2 ] ) , multiplying by and rearranging terms , we get ( [ eq : g2 ] ) .now set so that ( [ eq : g2 ] ) is equivalent to .observe that and , by ( [ eq : alpha ] ) , that where we used ( [ eq : t0:sinh : cosh ] ) with to get the last equality .looking back to ( [ eq : htilde : tc+mu ] ) and ( [ eq : htilde : tc+mu : positive ] ) , we see that furthermore , from ( [ eq : def : hctilde : max ] ) , we see that the derivative of vanishes twice : at and at ( compare also with ( [ eq : def : mtilde ] ) ) . by ( [ eq : alpha ] ) , we have that since because by proposition [ prop : sol : fbp : min ] , we have that which is equivalent by the preceding to . therefore , the function is monotone decreasing on ,2\mu b_c[ ] which is the unique solution of ( [ eq : g2 ] ) .finally , we check that .this is clearly equivalent to showing that , where is the unique solution of on ,t_c+\mu[ ] .the function is strictly increasing on  ] .we have which is strictly negative on ,t_c+\mu[ ] , and from ( [ eq : symmetry : max ] ) .this completes the proof .now that we have the solution of the free boundary problem ( [ eq : fbp : wc ] ) , we shall prove that is equal to the value function and that is an optimal control .this will be similar to the proof of optimality in the minimization problem .the proof is the same as for proposition [ prop : esperance : tauac ] .[ lem : generator : wbar ] let denote the operator defined in ( [ eq : def : generator ] ) . then , for all , moreover , if , then .\ ] ]  the proof follows the same steps as for lemma [ lem : generator : vbar ] . in particular , for ^{\text{max}}_c , b^{\text{max}}_c [ \, ] , and so therefore , where we have used the equalities , , ( [ eq : alpha : bis ] ) and , and the fact that .this proves lemma [ lem : generator : wbar ] .[ lem : ineq : wbar ] for all and any , if , then   we compare the left- and right - hand sides of this inequality on a case by case basis as in the proof of lemma [ lem : ineq : vbar ] . in the casewhere , the result follows immediately from ( [ eq : sol : fbp : c*:w ] ) together with ( [ eq : ac*:argmax ] ) .assume and , without loss of generality , that ._ case 1 . _ if , then ^{\text{max}}_c,1] ] by .thus , is strictly decreasing on with and so for all . finally , if , then and if and only if ,1-b^{\text{max}}_c[ ] , then =d_1^{\text{max}} ] , then -a^{\text{max}}_c , a^{\text{max}}_c[ ] with , and by , we have that for all -a^{\text{max}}_c , a^{\text{max}}_c[\, ] , then and by , where is defined in .notice that by .it follows from the properties of that we already discussed in part 1 that for all ^{\text{max}}_c,1] ] by for , we consider the process .the next corollary follows immediately from lemma [ lem : ineq : wbar ] .[ cor : deltak ] let , let denote the control satisfying ( [ eq : def : candidate : gc ] ) and let be any admissible control .then , for all  ] , either or and , where : x^a_t\in c_{a_{t- } } \mbox { and } a_t\neq a_{t-}\right),\\   p_2&:= { \mathbb{p}}_{x , a}\left(\exists \ , t \in [ 0 , \tau^a [ : x^a_t\in d_{a_t}\right).\end{aligned}\ ] ] then is -sub - optimal , in the sense that   the condition means that with positive probability , the strategy prescribes at least once to switch drifts in the continuation area of the control .the condition means that with positive probability , the strategy prescribes at least once to continue without switching in a switching region of .if , then it is clear that .thus , we can assume that .suppose first that .then by corollary [ cor : deltah ] , setting in ( [ eq : inequality : itoformula : hca ] ) , we find that and taking the expectations , applying the monotone convergence and the dominated convergence theorems ( recall that is bounded ) , we get since the left - hand side is equal to , this proves the statement when .suppose now that .let denote lebesgue measure .notice that the interior int of is not empty for . by right - continuity of and because of the irregular behavior of sample paths of diffusion processes , on the event , setting in ( [ eq : equality : itoformula : hca ] ) and then applying successively lemma [ lem : generator : vbar ] and corollary [ cor : deltah ] , we get again by lemma [ lem : generator : vbar ] , the integrand of the last integral is strictly positive if belongs to the interior of .taking expectations in the previous inequality and applying successively the dominated and monotone convergence theorems and the hypothesis with , we find that which concludes the proof .if we consider a diffusion coefficient for the particle , we can deduce from theorem [ verification : thm : min ] the corresponding value function and optimal control .let and let be a standard brownian motion , be its natural filtration and be the associated set of strategies . for , consider the s.d.e . and the corresponding stochastic control problem whose value function is given by where is the exit time of from  ] and $ ] .thus , there exists only a weak solution of ( [ eq : eds : xa0 ] ) which is given by tanaka s formula ( see e.g. ( * ? ? ?* sections 7.3 and 10.4 ) ) .in the next proposition , we show that can be obtained as a limit of as .observe first that when , equations ( [ eq : bc ] ) and ( [ eq : ac ] ) become respectively where .the unique solution of the first equation is , so .the second equation has exactly two solutions , one of which is and the other is . since , we deduce that . putting these values into ( [ eq : alpha ] ) and ( [ eq : beta ] ) , respectively, we find that when , and .therefore , according to ( [ eq : sol : fbp : min : c < c * ] ) , we find that , \\[7pt ]                     \frac{1-x}{\mu}-\frac{1}{2\mu^2}e^{-\mu } \left(e^{2\mu(1-x)}-1\right ) , & x\in\,\left]\frac{1}{2},1\right ] .\\           \end{array}\right.\ ] ] these formulas coincide with ( [ eq : v : limit0 ] ) and the proof is complete .  in figure[ figure : comparison : wvf ] , we give the graph of the value function of both problems ( [ eq : control : problem : min ] ) and ( [ eq : control : problem : max ] ) for two possible values of the switching cost ( and ) when the intensity of the drift is . in this case , the critical value of the cost is .the numerical value of the switching boundaries in the case where are given by : and in the case where , they are given by : because of the symmetry , figure [ figure : comparison : wvf ] shows only the value function corresponding to a positive initial drift .   and for two different values of the cost .the dotted line is the graph of the function defined in ( [ eq : def : fnu ] ) , which coincides with the payoff of the constant strategy.,title="fig:",width=264 ]  and for two different values of the cost .the dotted line is the graph of the function defined in ( [ eq : def : fnu ] ) , which coincides with the payoff of the constant strategy.,title="fig:",width=264 ]

Abstract: we solve two stochastic control problems in which a player tries to minimize or maximize the exit time from an interval of a brownian particle , by controlling its drift . the player can change from one drift to another but is subject to a switching cost . in each problem , the value function is written as the solution of a free boundary problem involving second order ordinary differential equations , in which the unknown boundaries are found by applying the principle of smooth fit . for both problems , we compute the value function , we exhibit the optimal strategy and we prove its generic uniqueness .   * optimal expulsion and optimal confinement + of a brownian particle with a switching cost * and * laura vinckenbosch *  cole polytechnique fdrale de lausanne   * keywords : * stochastic control with switching cost , principle of smooth fit , free boundary problems , martingale method . * 2010 msc subject classifications * primary : 93e20 . secondary : 60g40 , 60j65 .


in recent years , a substantial amount of research has been focused on model systems of heterogeneous adaptive agents interacting competitively , as e.g. in games , markets or ecosystems , in the attempt to understand the mechanisms by which real systems create exploitable information , and to clarify the origin of their complex collective behavior .the minority game , with its several variants , is perhaps the most studied of such models . in its simplest version ,it describes a population of boundedly rational players with fully heterogeneous beliefs who , at each round of the game , make their strategic decisions basing on some public information pattern ( the `` state of the world '' ) aiming to be in the minority group .the minority - wins mechanism , which serves the purpose of modeling competition for a scarce resource , translates into a strong assumption on the behavioral traits and expectations of players .indeed , it turns out that in order to maximize their expected utilities under the minority - wins rule , agents have to enhance their initial heterogeneity and differentiate themselves as much as possible from each other .this is rather intuitive : if agents would learn to make decisions similarly to each other , being in a minority would become a rather unlikely event . on the other hand , one might also consider another tendency that is often encountered in real agents , namely that toward imitation , say of an agent who believes that his / her payoff is maximized when he / she acts according to the majority . in this paper , we consider a mixed majority - minority game , to study the effects of competition in a population formed by two types of players , i.e. those whose short - term behavior is driven by imitation ( who play a majority game ) , and those who are instead anti - imitative ( and play a minority game ) .  from the viewpoint of economic modeling , our system represents a simple abstraction for a market where two classes of economic agents , namely `` fundamentalists '' and `` trend - followers '' , interact . the former see for details create their expectations under the assumption that the market price is close to its `` fundamental '' value , i.e. to a stationary equilibrium , and correspond to minority game players .the latter , instead , extrapolate a trend from recent price increments and assume that the next increment will occur in the direction of the trend ( see also ) ; they correspond to majority game players . in real markets , fundamentalists act as a kind of elastic force that pulls the price toward its fundamental value , while trend - followers destabilize the market by driving the price away from it .they are in particular widely believed to be the main actors in the infamous buy rushes known as `` bubbles '' . understandably then, modeling the interplay between trend - followers and fundamentalists is a basic issue in the theory of markets , and several models have been proposed ( see e.g. and references therein ) . in most cases , however , insight can be gained only from numerical simulations due to the complexity of the microscopic definitions .the mixed model we consider here has the advantage of being simple enough to be analytically tractable via the methods of statistical mechanics , notwithstanding its phenomenological richness .the effects due to the presence of trend - followers are fully discernible and an interpretation in market terms is quite straightforward . besides , the majority game is an interesting model in itself , that from the theoretical viewpoint shares some features with the hopfield model .surprisingly , it has not received much attention so far .this work is organized as follows .the basic definitions of the model are given in sec . 2 , together with an outline of the results .the static approximation to the analysis of the asymptotic macroscopic properties is expounded in sec .it is based on the formal analogy with zero - temperature spin glasses first derived in for the pure minority game , whose stationary states were shown to be ( approximately ) given by the minima of a random hamiltonian . in our case , the resulting optimization problem is slightly more subtle and its solution requires a negative dimensional replica theory of the kind already used for `` minimax games '' , close in spirit to the method of partial annealing .sec . 4 is devoted to the dynamical solution of the `` batch '' version of the model , which is carried out employing the generating functional technique along the lines of .some details about this calculation are given in the appendix . finally , in sec .5 , we formulate our conclusions .the setup we consider is as follows .there are players and possible information patterns . for each player two strategies are given ( ) that map an information pattern into a binary trading action ( ` buy / sell ' ) .( the generalization to strategies per agent is possible but it is analytically less convenient . ) we assume as usual that scales with so that remains finite in the relevant limit and that each is selected randomly with uniform probability in at the beginning of the game for all , and and fixed .strategies are evaluated according to their `` performance '' . at each round , players receive an information pattern chosen at random with uniform probability in  .subsequently , each player picks his so - far best - performing strategy , , and formulates the bid it prescribes , i.e. . the aggregate action of all players at round ( in economic terms , the `` excess demand '' ) is just once is known , majority ( resp .minority ) game players reward their strategies for which ( resp . ) . hence the performance updating or learning process takes place according to where for minority game players and for majority game players , and the game moves into the next round .the s can be seen as an additional family of quenched r.v.s ( besides the s ) with probability density .  for later use , it is convenient to introduce the `` preferences '' and the quantities , and , using which ( [ pud ] ) can be recast as an equation for : \ ] ] where  ] ) .however , they are the simplest possible in absence of retrieval states. we will adopt them for this reason , but it should be kept in mind that they may not be the most appropriate ones in general .after some algebra , the set of saddle point equations can be greatly simplified , because , as in , when the averages ( [ ave1 ] ) and ( [ ave2 ] ) can be explicitly performed by steepest descent .the result for the relevant quantities is with } ] and is a time - dependent external field . in principle ,disorder - averaged correlation and response functions can be calculated exactly at all times by taking appropriate derivatives of the disorder - averaged , i.e. }}=\int e^{{{\rm i}}\sum_{it}\widehat{y}_i(t){\left}[y_i(t+1)-y_i(t)-\theta_i(t){\right}]+ y_i(t)\psi_i(t)+n f(\boldsymbol{\widehat{y } } ) } p(\boldsymbol{y}(0 ) ) d(\boldsymbol{y},\boldsymbol{\widehat{y}})\\ f(\boldsymbol{\widehat{y}})=\frac{1}{n}\log{\overline{{\left}[e^{-{{\rm i}}\sum_{it } \widehat{y}_i(t)\epsilon_i{\left}[h_i+\sum_j j_{ij}s_j(t){\right}]}{\right}]}}\end{aligned}\ ] ] with respect to the fields and .we shall however be interested in the stationary state only .as usual , evaluation of leads to an effective ( non - markovian ) process that provides an equivalent description of the original ( markovian ) multi - agent process ( [ batch ] ) .such a calculation is in this case rather similar to that done for the pure batch minority game in , and is sketched in the appendix .the main difference is that here we obtain _ two _ effective processes , describing trend - followers and fundamentalists respectively .these are given by {tt'}s(t')+\theta(t)+ \sqrt{\alpha}z(t)\ ] ] where ( resp . ) for the majority ( resp . minority )part , and is a zero - average gaussian random variable with temporal correlations {tt'}\ ] ] stands for the identity matrix while denotes the matrix with all elements equal to one . has elements . , instead ( see appendix for details ) , is the sum of two contributions :  ( resp . ) has elements ( resp . ) where the subscript means average over the process ( [ esap ] ) with ( resp . ) . when , can be identified with the disorder- and agent - averaged autocorrelation function of ( [ batch ] ) , while the two components of become identical to the disorder- and agent - averaged response functions of minority and majority agents , respectively .ergodic stationary states can be studied under the following assumptions :  * time - translation invariance ( tti ) : ; * finite integrated ` response ' ( fir ) : ; * weak long - term memory ( wltm ) : finite .  the breakdown of any of these signals the breakdown of ergodicity . to be more clear, we remark that the ` integrated response ' defined in fir has two components , i.e. , with obvious notation , and can be negative . and are the actual susceptibilities of minority and majority agents , respectively . with fir , we will require that _ both _ and are finite .as in the minority game , for individual agents there are two possibilities : either as , in which case they use only one of their strategies asymptotically ( we call these agents `` frozen '' ) ; or as , in which case they keep flipping between their strategies even in the long run ( we call these agents `` fickle '' ). macroscopic quantities can be obtained by separating the contributions of frozen and fickle agents .defining ,  ] ) : +f\nonumber\\ & = ( 1-f){\left}[1-{{\rm erf}}\frac{\lambda}{\sqrt{2}}+\frac{1}{\lambda^2 } { \left}({{\rm erf}}\frac{\lambda}{\sqrt{2}}-\lambda\sqrt{\frac{2}{\pi } } e^{-\lambda^2/2}{\right}){\right}]+f\end{aligned}\ ] ] where .this agrees with the replica result ( [ crep ] ) .for the fraction of frozen agents one obtains in fig .[ cphi ] analytical results for and are compared with simulations .the ` susceptibility ' ( [ chiii ] ) can instead be calculated from the formula which follows directly from the fact that response functions for minority ( resp .majority ) agents can be obtained as ) in the same way , apart from the factor . ]/\partial z(t')}\right\rangle}_{-1} ] ) , after an integration by parts and a time average .the minority part is as usual given by to calculate the majority part , one must fix the value of the product for , where can be either or ( for one has in any case ) . in principle , there are several possibilities .if one makes the ` natural ' choice , then this leads to  diverges ( hence fir is violated and ergodicity is broken ) when the fraction of fickle agents satisfies or , equivalently , at the critical values of given by the equation where is the solution of ( [ chi1][cx ] ) are in full agreement with the replica results of sec .3 .  another possibility is to calculate without making any special assumption on for .this brings us to a situation where ( [ sz1][alfacr1 ] ) are substituted respectively by where now solves the value of at which diverges is now .notice that the extra exponential factor one obtains in this way does not change numerical results for significantly ( the solution of ( [ cx ] ) is in fact , so is always close to ) .notice also that for a purely majority game ( recalling that ) one gets for the susceptibility instead of the hopfield - like formula ( [ hoppp ] ) . in both cases , when .for the stationary volatility , which reads {tt}\ ] ] one can use the approximate method of to derive an expression in terms of the persistent parameters and , which holds for : solving for , and for different and substituting one obtains the volatility branches displayed in fig .[ figs2h ] , which are again in excellent agreement with simulations .to summarize , we have studied the mixed majority - minority game with random external information . neglecting ` retrieval ' ( i.e. the possibility that trend - followers flock in presence of a particular information pattern ), we have first calculated the stationary state of the dynamics from a static approximation via a negative - replica theory .then we have solved the dynamics using generating functional methods .the two approaches match nicely and agree with numerical results for the macroscopic observables and in a satisfactory way .this suggests that retrieval does not affect such quantities significantly .our results also indicate that when fundamentalists outnumber trend - followers , the macroscopic behavior of the system ( ` phase transition ' with ergodicity breaking from an inefficient phase at high to an efficient one at low ) can be explained by the onset of anomalous response , that is by a divergence of the integrated response , as in the pure minority game .we have calculated the line of critical points in the plane showing that the inefficient phase gets larger as increases .when trend - followers dominate , instead , the system is always inefficient and low volatility states disappear . as a byproduct, we have provided an approximate static and dynamical solution of the majority game .a greater effort is nevertheless needed in order to incorporate the possibility of ` herding ' in both the replica theory and the path - integral solution .we expect retrieval states to exist at low for any . while such states should nt affect global time - averaged properties ( i.e. and ) significantly , they are likely to play a most crucial role in such phenomena as `` bubbles '' , that in our setting can be seen as localized in time .it is also likely that rsb occurs at very low for any , in pretty much the same way as rsb occurs for any non - zero market impact in the pure minority game .let us finally remark two aspects of the present model that can be criticized and hence improved . in first place , all players can in principlewin at the same time ( i.e. the available resources are infinite ) , which is a clearly unrealistic situation ( albeit extremely unlikely in our disordered setup with ) . secondly , in a market a large buy rush today is justified by the belief that _ tomorrow _ the price will rise again so that for instance one will be able to sell at a higher price .so in a majority game it would perhaps be more correct to measure the effectiveness of a trading decision made today by what the payoff will have been tomorrow .in other words , a player making a trading decision at round should receive a payoff at round .instead , in our model , his payoff is . in spite of these limitations, we see that our model does indeed capture some of the features one expects to find in markets where fundamentalists and trend - followers compete .also , we believe that some of the issues listed above , starting with retrieval , can be taken into account , possibly with modest modifications . in our view , a possibly more interesting generalization would consist in allowing the s to be dynamical variables , in order to give agents the possibility to change their character .some work along these lines is currently in progress .we are grateful to a cavagna , m marsili , g parisi and f ricci - tersenghi for useful discussions and suggestions , and in particular to m marsili for disclosing some results of prior to publication .we would also like to thank acc coolen for introducing us to dynamical methods .the disorder average is as usual expected to generate two - time player - averaged functions of the and variables only .we focus on the matrix can be seen as formed by two components , for minority and majority agents , respectively : forcing the above definitions inside via -functions with the proper -scaling and assuming that , we find ( with the shorthand ) }}=\int e^{n(\psi+\omega+\phi ) } d(\mathsf{q},\mathsf{\widehat{q } } ) d(\mathsf{l},\mathsf{\widehat{l } } ) d(\mathsf{k},\mathsf{\widehat{k}})\ ] ] where  ] comes from the saddle point described by the equations where with + y(t)\psi_i(t)}\times\nonumber\\ \times e^{-{{\rm i}}\sum_{tt'}[s(t)\widehat{q}_{tt'}s(t')+\widehat{y}(t ) \widehat{l}_{tt ' } \widehat{y}(t')-\epsilon_i s(t)\widehat{k}_{tt'}\widehat{y}(t')]}\end{aligned}\ ] ] it can be checked by a direct calculation ( e.g. following ) that , at the relevant saddle point , as for , one can define and see , for instance by taking the derivative of with respect to , that where is the response function of minority agents , with elements and similarly is the response function of majority agents .  setting the generating field to zero and assuming that , we can now treat minority agents ( ) and majority agents ( ) separately .we get , for : }\times\nonumber\\ \times e^{-{{\rm i}}\sum_{tt'}[s(t)\widehat{c}_{tt'}s(t')+\widehat{y}(t ) \widehat{l}_{tt ' } \widehat{y}(t')-\epsilon s(t)\widehat{k}_{tt'}\widehat{y}(t')]}p(y(0 ) ) d(y,\widehat{y})\end{aligned}\ ] ] where we set ; the measure instead becomes }\end{aligned}\ ] ] and represent majority and minority agents , respectively .the saddle - point equations for , and are identical to those found for the pure batch minority game .it results that {tt'}\\ \widehat{l}_{tt'}=-\frac{1}{2}{{\rm i}}\alpha[(\mathsf{i}-{{\rm i}}\mathsf{k})^{-1 } ( \mathsf{e}+\mathsf{c})(\mathsf{i}-{{\rm i}}\mathsf{k}^t)^{-1}]_{tt'}\nonumber\end{aligned}\ ] ] where and . substituting these into one obtains {tt'}\widehat{y}(t')}\times\nonumber\\\timese^{{{\rm i}}\sum_t\widehat{y}(t ) [ y(t+1)-y(t)-\theta(t)-\alpha\epsilon\sum_{t'}[(\mathsf{i}-{{\rm i}}\mathsf{k})^{-1}]_{tt ' } s(t)]}\end{aligned}\ ] ] recalling that , it turns out that the disorder - averaged correlation and response functions for minority and majority agents are obtained as averages over the colored effective stochastic processes ( [ esap ] ) with and , respectively .

Abstract: we study the asymptotic macroscopic properties of the mixed majority - minority game , modeling a population in which two types of heterogeneous adaptive agents , namely `` fundamentalists '' driven by differentiation and `` trend - followers '' driven by imitation , interact . the presence of a fraction of trend - followers is shown to induce ( a ) a significant loss of informational efficiency with respect to a pure minority game ( in particular , an efficient , unpredictable phase exists only for ) , and ( b ) a catastrophic increase of global fluctuations for . we solve the model by means of an approximate static ( replica ) theory and by a direct dynamical ( generating functional ) technique . the two approaches coincide and match numerical results convincingly .