let us denote the norm and the inner product in , i.e. ,    in this paper , we consider the following problem : determine a function \times\left[0,t\right]\to l^{2}\left(0,\pi\right) ] ;  ( a2 ) ;\mathcal{d}\left(\mathcal{a}\right)\right)\cap c^{1}\left(\left[0,t\right];\mathcal{h}\right) ] .  in the following theorems ,we show the formula of solution of the problem ( [ eq:2 ] ) by employing theorem 1.1 in with and following the steps in .under the conditions ( a1)-(a4 ) , the problem  \times\left[0,t\right],\\ u\left(0,s\right)=\psi\left(s\right ) , & s\in\left[0,t\right],\\ u\left(t,0\right)=\varphi\left(t\right ) , & t\in\left[0,t\right ] , \end{cases}\label{eq:4}\ ] ]  admits a unique solution presented by the following formula . for any ,    and for any ,    moreover, the solution belongs to the space \times\left[0,s\right];\mathcal{d}\left(\mathcal{a}\right)\right)\cap c^{1}\left(\left[0,t\right]\times\left[0,s\right];\mathcal{h}\right) ] satisfies the problem ( [ eq:4 ] ) , namely ,  \times\left[0,t\right],\\ v\left(0,\xi\right)=\psi_{1}\left(\xi\right)\equiv u\left(t , t-\xi\right ) , & \xi\in\left[0,t\right],\\ v\left(\tau,0\right)=\varphi_{1}\left(\tau\right)\equiv u\left(t-\tau , t\right ) , & \tau\in\left[0,t\right ] .\end{cases}\ ] ]  thanks to theorem 2 , is given by    it follows that    thus , we obtain    by the maps in the integrals .we can see by the initial conditions of ( [ eq:5 ] ) that      which leads to    by virtual of semi - group properties , we get    substituting ( [ eq:7 ] ) into ( [ eq:6 ] ) , we thus have    under the conditions ( a1 ) , ( a2 ) and ( a4 ) , if the problem ( [ eq:2 ] ) with admits a solution , then this solution can be given by    now we put and , then write    the function \times\left[0,t\right]\to\mathcal{h} ] are given . in this problem, we have and , so there exists an orthonormal basis of , satisfying ( see e.g. )  \right),\quad\delta\phi_{n}=-\lambda_{n}\phi_{n},\quad0<\lambda_{1}\le\lambda_{2}\le ... \lim_{n\to\infty}\lambda_{n}=\infty.\ ] ]  the laplace operator thus has a discrete spectrum with and gives the orthonormal eigenbasis for .then , thanks to those theorems above , the solution has the form    where    we can see that the instability is caused by all of the exponential functions .in fact , let us see the case in ( [ eq:8 ] ) . since the discrete spectrum increases monotonically as tends to infinity , the rapid escalation of and is mainly the instability cause . even though these exact given functions may tend to zero very fast , performing classical calculation is impossible .it is because that the given data may be diffused by a variety of reasons such as round - off errors , measurement errors .a small perturbation in the data can arbitrarily generate a large error in the solution . a regularization method is thus required .in this section , assuming that the problem has an exact solution satisfying various corresponding assumptions , we construct the regularized solution depending continuously on the data such that converges to the exact solution in some sense .moreover , the accuracy of regularized solution is estimated .  the solution of ( [ eq:1 ] )can be given by    we shall replace all instability terms by the better ones , particularly , and where is a real number . then , the regularized solution corresponding to the exact data is    for any , and    for any .we also denote the regularized solution corresponding to the perturbed data by    for any , and    for any .now we shall show two elementary inequalities in the following lemmas .  for ,we have    it is obvious that since .  for all , we have    the proof of this lemma is based on the fact that .therefore , we have  ^{\frac{1}{\alpha}},\end{aligned}\ ] ]  which leads to    in the sequel , we only prove the case in our main result because of the similarity .the results are about the regularized solution depending continuously on the corresponding data and the convergence of that solution to the exact solution .now we shall use two elementary lemmas above to support the proof of the main results .  under the conditions ( a1 ) , ( a2 ) , ( a4 ) andassume that , then the function given by ( [ eq:10])-([eq:11 - 1 ] ) depends continuously on in .let and be two solutions of ( [ eq:10])-([eq:11 - 1 ] ) corresponding to the data and , respectively . by using parseval relation , for have    similarly , for any , we get    under the conditions ( a1 ) , ( a2 ) and ( a4 ) , if the problem ( [ eq:1 ] ) with admits a unique solution satisfying    and    where , let be perturbed functions satisfying the conditions ( a1)-(a2 ) , respectively , and let be the regularized solution , given by ( [ eq:11])-([eq:12 ] ) , corresponding to the perturbed data , then for we have    and for ,    for any , we have        using triangle inequality , in order to get the error estimate , we have to estimate and .indeed , we get    next , can be estimated as follows .we put    then we have    therefore , we conclude that    now using parseval relation again , we thus obtain    thanks to lemma 6 and the assumption ( [ eq:14 - 1 ] ) , we have    combining ( [ eq:16])-([eq:17 ] ) , we obtain    similarly , we obtain the error estimate    for the case with the assumption ( [ eq:15 ] ) .  hence , we complete the proof .  from theorem 8, we can see that strongly converges to in for any \times\left[0,t\right] ] , for example , and nearly have the same rate of convergence in some cases .indeed , the convergence speed at is and it is of order for .then , if is very large for any fixed , the order may approach .this creates the globally stability behavior of the error in numerical sense . on the other hand , the natural acceptance of ( [ eq:14 - 1])-([eq:15 ] )can be obtained at .namely , by letting the conditions become    moreover , the error is of order for all \times\left[0,t\right] ] . for these given functions ,the problem has a unique solution    now let us take perturbation on data functions as follows . for , we define      thus , the solution corresponding to the perturbed data functions is    it is easy to see that converges to over the norm as . to observe the ill - posedness ,we can compute , for example , and    therefore , we get    as .this divergence is also showed in figure 1 with and .   and the approximation solution within perturbations . ]  .comparison of absolute errors between the regularized solutions of and .[ cols="^,^,^,^,^,^ " , ]   . ]now we compute the regularized solution based on ( [ eq:11])-([eq:12 ] ) as follows .    for any , and    for any .  to obtain numerical results, we use a uniform grid of mesh - points where    we thus seek the discrete solutions and given by ( [ eq:13 - 1 ] ) and ( [ eq:13])-([eq:14 ] ) , respectively .  by fixing and ,the numerical results are shown in table 1 and illustrated in figs .2 - 3 as below .2 is the graphical representations for curved surfaces of the exact solution , and of the approximate solution determined in ( [ eq:13])-([eq:14 ] ) with . in fig .3 , we have drawn the exact solution and the approximate solution where are , and , respectively , in order to see the convergence at as becomes very large , namely , the bound in theoretical result tends to zero . as in figs . 2 - 3, we can conclude that the regularized solution converges to the exact one as the error becomes smaller and smaller .moreover , convergence is , particularly , observed from the absolute ( abs . )errors in table 1 .hence , our numerical results are all reasonable for the theoretical result .in this work , a regularization method has been successfully applied to the inverse ultraparabolic problem .this method is to replace the instability terms appearing in the formula of the solution which is employed by semi - group theory .therefore , such a way forms the so - called regularized solution which strongly converges to the exact solution in -norm .we also obtain the error estimate which is of order . by a numerical example , application of the method is flexible and calculation of successive approximations is direct and straightforward .this work is more general than , a recent work of zouyed et al ., in both error estimate and the considered problem .the authors declare that they have no competing interests .vak , ltl organized and wrote this manuscript .vak , ltl and tth contributed to all the steps of the proofs in this research together .nht participated in the discussion and corrected the main results .all authors read and approved the final manuscript .the authors wish to express their sincere thanks to the anonymous referees and the handling editor for many constructive comments leading to the improved version of this paper .a. i. kozhanov , on the solvability of boundary value problems for quasilinear ultraparabolic equations in some mathematical models of the dynamics of biological systems , journal of applied and industrial mathematics , 2010 , vol .512 - 525 .v. s. dron and s. d. ivasyshen , properties of the fundamental solutions and uniqueness theorems for the solutions of the cauchy problem for one class of ultraparabolic equations , ukrainian mathematical journal , vol .50 , no . 11 , 1998

Abstract: in this paper , we study the inverse problem for a class of abstract ultraparabolic equations which is well - known to be ill - posed . we employ some elementary results of semi - group theory to present the formula of solution , then show the instability cause . since the solution exhibits unstable dependence on the given data functions , we propose a new regularization method to stabilize the solution . then obtain the error estimate . a numerical example shows that the method is efficient and feasible . this work slightly extends to the earlier results in zouyed et al .  ( 2014 ) .


since its invention , atomic force microscopy ( afm ) has become a routine technique for surface characterization at high spatial resolution . of the various modes of afm operation, dynamic - mode afm offers the advantage of reduced drift effect and reduced lateral drag forces on the sample , thus greatly contributing to the versatility of the technique . in dynamic - mode afm ,the cantilever is oscillated at its resonance frequency and the interaction with the sample is probed via changes in the cantilever oscillation .this is significantly more complicated in liquid , because of viscous damping . in aqueous solution , the natural environment for biological molecules ,cantilever quality factors are of order one .such low quality factors imply a less harmonic and thus harder to interpret cantilever response , yield reduced force sensitivity and , for stiffer cantilevers , an increased relative importance of the noise in the detection of the cantilever deflection .these problems have at least partly been overcome in specialized equipment , by the development of low - noise deflection detectors and operation at small oscillation amplitudes , resulting among others in atomic resolution on solid - liquid interfaces by frequency - modulation afm . in an alternative , `` diving - bell '' approach ,only a small portion of the afm tip was dipped in the liquid medium , leading to significantly higher quality factors than when the whole cantilever is submerged .the problems persist , however , in many standard afm systems .an even more general problem lies in the actuation of the cantilever resonance .a standard piezo - acoustic drive actuates the whole fluid cell or at best the cantilever support chip , and simultaneously excites a wide range of mechanical resonances , commonly known as the `` forest of peaks '' that is observed in liquid afm .such spurious mechanical resonances usually dominate the low- resonance of the cantilever itself , which makes it hard to tune to the cantilever resonance or specifically enhance its effective by external positive feedback and which affects any quantitative interpretation of phase contrast of measured frequency shifts .in addition , because of the macroscopic nature of these spurious resonances , they are prone to drift .these problems have motivated a wide variety of approaches to uniquely actuate the cantilever resonance , i.e. , without actuating any other , spurious resonances .the cantilever response should thus closely resemble that of a simple harmonic oscillator .while progress has been made by optimizing piezo - acoustic actuation , it has been suggested that the focus in the future should be on `` direct '' actuation methods .magnetic , photothermal and electrostatic actuation provide valuable alternatives that , unlike piezo - acoustic methods , only excite the cantilever itself .  in this work ,we have attempted to overcome the problems related to viscous damping by enhancing the ( intrinsic ) cantilever quality factor .to do so , we deliberately increased the inertial mass of the cantilever by attaching beads with tens of m diameter and g masses . while there are models to determine the shift in the resonance frequency after the attachment of beads at a specific position on the cantilever , these models are only valid for a bead mass much smaller than the cantilever mass .however , to achieve a significant increase in the quality factor , beads of masses greater than the cantilever mass are needed .we chose magnetic beads to facilitate the comparison between piezo - acoustic actuation and `` ideal '' magnetic actuation of the cantilevers .  in this paper , we first describe the experimental setup for magnetic actuation and bead attachment ( section [ sec : expsetup ] ) .next ( section [ sec : results ] ) , we demonstrate the effect of the beads on the thermal noise spectrum and resonant response of the cantilever to external actuation , and provide an analysis of the minimum force gradient that can be detected by cantilevers with beads of various sizes .we also include a demonstration of afm imaging in liquid using this method .we summarize our results in section [ sec : comment ] , before giving further details on the magnetic actuation in appendix [ sec : solenoid ] .we expect such details to be of wider applicability and use for implemention in any afm system that is mounted on an inverted optical microscope .all experiments and operations were carried out with a nanowizard i afm system ( jpk instruments , berlin , germany ) , mounted on an inverted light microscope ( ix71 , olympus , tokio , japan ) .results will be shown / discussed for ppp - nch and ppp - fm silicon cantilevers ( nanosensors , neuchtel , switzerland ) .the nominal dimensions ( length width thickness ) are and for the nch and fm cantilevers , respectively .cantilever deflections were measured with conventional beam deflection , and the detector was calibrated by pressing the cantilever into a hard substrate by ramping the ( calibrated ) scanner .beads were selected from an isotropic magnetic powder ( mqp - s-11 - 9 , magnequench , now part of : molycorp , greenwood village , colorado , usa ) .they were selected for their size as measured on the inverted optical microscope , using a calibrated grid on the output of the camera on the microscope .bead masses were estimated based on their diameter and the mass density of the bead material .  of the various methods to attach particlesto cantilevers , we chose the combination of the afm head and inverted light microscope as a micromanipulator for the attachment process , and used a uv - curing glue ( glass bond , loctite ) as an adhesive . to optimize the process of bead attachment , a customized , perspex cantilever holder was used , see fig .[ fig : bead]a - b .it partly follows the design of the standard cantilever holder / fluid cell and is accordingly fixed in the afm system , but enables us to keep the cantilever at an optimum angle to pick up the adhesive and the bead without risking to touch the underlying glass slide with the support chip .the cantilever was ( temporarily ) fixed in the customized cantilever holder with blu - tack in such a way that the back side of the cantilever faces the sample holder .( color online . ) ( a ) photograph of the home - made cantilever holder designed for attaching beads using the afm head and the inverted optical microscope .( b ) schematic of the cantilever mounted on the block approaching a bead on a glass slide ( not to scale ) .( c ) optical microscope image of a bead on a ppp - nch cantilever . ]  to attach the beads , small portions of the adhesive and of the magnetic powder were placed on a glass slide , and the stepper motor of the afm was used to approach the cantilever to the glass surface .the process was monitored using the optical microscope and the step size of the motor was decreased as the cantilever got closer to the glass slide . after the back side of the cantilever touched the adhesive ,the cantilever was retracted .this procedure was repeated for picking up and attaching the bead .it yielded cantilevers with beads as shown in fig .[ fig : bead]c .the ideal resonant response of the cantilever was characterized by their thermal noise spectrum .cantilever resonance frequency , quality factor , and spring constant were determined from simple harmonic - oscillator fits to the thermal noise power . for the gold - coated ppp - nch cantilevers , the microscope had a noise floor of about 0.2 pm/ for detecting cantilever deflections at frequencies  khz , and 0.3 pm/ at frequencies  khz , where heterodyne mixing ( heterodyne thermal noise module , jpk instruments , berlin , germany ) was required to bring the noise down into the accessible frequency range of the controller and where the noise was increased due to aliasing from higher frequencies .all measurements were carried out in milliq water unless otherwise specified .thermal noise curves were recorded before and after attaching a bead at the end of a cantilever . to measure the effect of different beads on the same cantilever , each bead was removed after the measurements by subsequent immersions in isopropanol and water , before drying and attaching another .  for magnetic actuation , the beads on the cantilevers were magnetized along their axes by bringing them close to a small , strong ( grade : n42 ) permanent neodymium magnet ( f346 , magnet expert , sutton - on - trent , uk ) . the equivalent field at a distance of 2 mm is approximately 1.4 tesla . compared to other methods of preparing magnetized cantilevers ,magnetic beads have the advantage of enabling significant ( at least several nm ) oscillation amplitudes even for cantilevers with spring constants of  n / m .  to generate a driving magnetic field ,a solenoid was designed , including ferrite core , to fit into one of the objective threads of the inverted optical microscope ( appendix [ sec : solenoid ] ) .this arrangement , with a solenoid under the sample , is similar to the one adopted elsewhere . unlike designs where the cantilever is placed inside an electromagnet on the fluid cell , this has the advantage of facilitating heat dissipation in the solenoid away from the cantilever and sample .this arrangement implies that afm imaging with magnetic actuation can not be performed simultaneously with optical imaging using the inverted microscope , but it is straightforward to switch to optical microscopy in between afm scans by moving an objective to the optical axis instead of the solenoid .( color online . )thermal noise power spectra in water , for the same ppp - nch cantilever without ( right ) and with ( left ) an attached bead of 55 m diameter .the solid and dashed lines denote fits with a simple harmonic oscillator model in addition to a constant background noise .the background noise in the right part of the figure is larger because heterodyne mixing was required for the measurement at higher frequencies . ][ fig : comparison2 ] shows the thermal noise data for a ppp - nch cantilever without and with a bead of a diameter  m. not unsurprisingly , we observe a large drop in the cantilever resonance frequency on the attachment of such a large ( ca .0.5 g ) mass to the end of the cantilever ( ca .35 ng , based on the cantilever dimensions and the mass density of silicon ) . besides the drop of the resonance frequency , we also find a significant increase of the quality factor . as a consequence of this increase in ,the resonance of the cantilever clearly emerges from the noise of the deflection detector , whereas the resonance is hardly visible without a bead attached to the cantilever ( which can only partly be attributed to the higher noise in the upper frequency range , as explained in section [ sec : expsetup ] ) .thermal noise of cantilevers with beads of different sizes .an offset was added for each set of data for clarity .horizontal lines denote the zero line for each curve .all measurements were done with the same cantilever except for the bead with = 20 m , where another cantilever of the same wafer was used , with a matching resonance frequency ( 116 khz ) for the unloaded cantilever in water . ]  for a quantification of these effects , this experiment was repeated with different bead sizes , each time recording the thermal noise ( fig .[ fig : beadsize ] ) . , , and followed from fits to the thermal noise spectrum and are displayed in table [ tab : beadsizes ] .the bead mass ( ) was estimated by assuming them to be perfect spheres and based on a mass density of 7.43 g/ ( value provided by the supplier ) ..[tab : beadsizes ] effect of beads of different sizes on the dynamic properties of an ppp - nch cantilever , corresponding to the data displayed in fig .[ fig : beadsize ] . [ cols="^,^,^,^,^",options="header " , ]   these results confirm the drop in resonance frequency , by up to a factor 5 , and the increase in quality factor of nearly a factor 3 for the larger beads , up to values of .this is to be compared to that is measured here and elsewhere for these cantilevers without beads attached .the variations in spring constant can be attributed to the uncertainty in the fits and/or to the adhesive for attaching the different beads .similar effects were observed for the softer ppp - fm cantilevers ( spring constant for the used unloaded cantilever :  n / m ) , where the resonance frequency dropped from 25.8 to 8.11 khz while the q - factor increased from to after attaching a bead with a diameter of 50 m.   ( color online . ) ( a ) the effective mass of the cantilever as a function of the bead mass .the straight line is a linear fit with the constraint of positive values for zero bead mass .( b ) damping coefficient as a function of bead radius .the dashed line indicates stokes s law . ]to further interpret these results , we calculated the effective cantilever mass , which would be the mass in an idealized mass - on - a - spring model of the cantilever ( where ) .[ fig : effectmass]a shows the as a function of the actual bead mass . scales linearly with the bead mass ( within the experimental errors ) . especially for the larger beads , , indicating that the dominant contribution to the cantilever inertia is from the bead , and not from the displaced liquid or the cantilever itself .the fact that may be attributed to the fact that the center of the gravity of the bead is not located at the very end of the cantilever . within the same model , we can assume a viscous damping force of the form , with the damping coefficient and the velocity of the bead .plotted as a function of bead radius ( fig .[ fig : effectmass]b ) , hardly increases until the bead size becomes significantly larger than the cantilever diameter .this demonstrates that does not show significant frequency dependence over the range explored in our experiments .the results for can be compared to stokes s law ( , with the viscosity of water ) , which is valid for a sphere in the limit of a low reynolds number , where the inertial mass becomes irrelevant .higher values , as found here , are convenient for finding and tracking the cantilever resonance .the value is also an important factor , however , for determining the force sensitivity , or more strictly speaking , the minimum detectable force gradient in the proximity of the tip to the sample in an afm experiment . to estimate this quantity ,we adopt a commonly used approach : we focus on elastic interactions only and assume that the force gradient is constant over the whole cantilever oscillation .a force gradient will then cause a well - defined shift of the cantilever resonance frequency . under the assumption that this shift is small compared to the width of the resonance , as is appropriate in liquid , this implies a change in cantilever phase with respect to the phase of a free cantilever oscillation that is driven near the resonance frequency .since the phase noise of an oscillation is directly related to the amplitude noise via , the minimum detectable force gradient therefore follows from with the amplitude of the oscillation ( all amplitudes are rms values unless stated otherwise ) .in a general afm experiment , the amplitude noise is the vectorial sum of the thermal cantilever noise and the amplitude noise caused by the deflection detector , . if the detector noise is negligible , this can be shown via application of the equipartition theorem to yield the well - known result measured in nn / nm/ , with the damping coefficient as defined earlier , and where is the boltzmann constant and the absolute temperature .the expression with is a representation of the fluctuation - dissipation theorem ( see , e.g. , ref . ) , which states that the thermal force noise power is proportional to the damping and not affected by the inertial mass . since does not show significant frequency dependence over the range explored in our experiments ( fig .[ fig : effectmass]b ) , it follows that the increase in inertial mass of the cantilever does not affect the force sensitivity at resonance .( color online . )the minimum detectable force gradient for an oscillating cantilever as a function of the mass of an attached bead , multiplied by the oscillation amplitude .data are shown for the ideal case of negligible detector noise ( eq . [ eq : minforcegrad_thermal ] ) and for the force gradient based on the measured amplitude noise at resonance ( eq . [ eq : minforcegrad ] ) .in addition , for bead masses  g , data are also shown based on the amplitude noise that has been corrected for the additional detector noise that was caused by the heterodyne mixing for the measurements at higher frequencies . ]this can be made more explicit by plotting in fig .[ fig : minforce ] , both for the ideal case of negligible detector noise ( eq . [ eq : minforcegrad_thermal ] , using the measured , , and ) and based on the measured amplitude noise ( eq . [ eq : minforcegrad ] ) as observed in fig .[ fig : beadsize ] .the amplitude noise at resonance was obtained from the parameters of the fit of the harmonic oscillator model plus the offset that corresponds to the detector noise . for any can be obtained by dividing the values in fig .[ fig : minforce ] by the oscillation amplitude . without detector noise ( eq .[ eq : minforcegrad_thermal ] ) , we find that the addition of a g bead to a cantilever of 35 ng hardly affects its force sensitivity .the force sensitivity only shows a significant , though moderate degradation for beads that are significantly larger than the cantilever width ( see the increase in for bead masses  g , in fig .[ fig : minforce ] ) , correlated with the increase in damping in fig .[ fig : effectmass]b .( color online . ) ( a ) resonant response of an ppp - nch cantilever without bead and with bead ( diameter ca .55 m ) , compared to magnetic actuation of the same cantilever ( with the bead ) .( b ) the response near the resonance , for the cantilever with bead , for both piezo - acoustic ( dotted line ) and magnetic actuation ( solid line ) , overlayed with the thermal noise ( scaled and referred to the right axis ) .( c ) as ( b ) , for the phase response , compared to the theoretical phase response ( dashed line ) expected from a simple - harmonic - oscillator fit to the thermal noise . ]  in general , eq .[ eq : minforcegrad_thermal ] shows that a drop in resonance frequency will not significantly affect if it is accompanied by a similarly sized increase in quality factor ( assuming a constant spring constant ) .the results in table [ tab : beadsizes ] and fig .[ fig : minforce ] thus demonstrate that the attachment of a larger bead does not significantly affect the sensitivity in an afm experiment . at the same time, it has the advantage of a sharper and thus more easily detectable cantilever resonance .importantly , fig .[ fig : minforce ] also shows that the attachment of such beads can _ improve _ the force sensitivity ( i.e. , minimize ) if the detector noise is significant .the attachment of beads can thus be a valuable method to overcome the detector noise in any afm system , without any adjustments to the microscope itself .in these experiments , a 50 m , 0.5 g bead yields the lowest noise ( fig . [fig : minforce ] ) combined with the highest ( ) quality factor ( table [ tab : beadsizes ] ) .  to demonstrate the practical advantage of attaching beads to cantilevers , we actuated the ppp - nch cantilever without and with bead by standard piezo - acoustic actuation , and compared the resulting resonant response to the result obtained ( with bead ) by magnetic actuation , see fig .[ fig : comparison]a . without bead , the piezo - acoustic actuation yields a broad `` forest of peaks '' , as fluid - borne vibrations lead to additional spurious resonances which obscure the cantilever resonance and cause deviations from the theoretically expected frequency response of the cantilever . with bead ,the spurious resonances are still very noticeable , but the higher causes the cantilever resonance to stand out against the spurious background , and at resonance ( 30 khz ) , the response is comparable to the ideal , magnetically actuated response ( fig . [fig : comparison]b - c ) . for the magnetic actuation ,the fluctuations below 20 khz are similar to earlier observations and may be attributed to mechanical vibrations of the coil in this frequency regime .( color online . ) ( a ) afm topography of 220 base - pair dna fragments .the cantilever was driven at its resonance of 29.5 khz with a peak amplitude of about 2 nm .the amplitude setpoint was optimized for best image contrast and stability .the raw data was modified by first - order line - wise flattening and gaussian filtering with  nm ( 1 pixel ) .( b ) average cross - sectional profile of the dna molecule indicated by the arrow . ]importantly , the amplitude and phase response around the resonance for both piezo - acoustic and magnetic actuation closely match the ideal response , as determined from the thermal noise spectrum fig .[ fig : comparison]b - c .we note that this is much less obvious for the first overtone of the cantilever ( with bead ) , which is observed at 150 khz , such that magnetic actuation would still be preferable if the attached bead is magnetic and a driving field can be generated .  finally , to illustrate the use of the cantilevers with beads , we imaged double - stranded dna fragments of 220 base pairs in buffer solution ( fig .[ fig : imaging ] ) , using standard intermittent - contact ( tapping/ amplitude - modulation ) afm and magnetic actuation of an ppp - nch cantilever with a 55 m diameter bead .the dna fragments were adsorbed on a mica substrate that was pre - treated by 2-minute incubation in 10 mm nicl-solution .the dna was adsorbed ( at a concentration of about 10 g / ml ) and imaged in 3 mm nicl , 10 mm hepes ph 7.0 aqueous solution .we found that the used afm set - up was too sensitive to acoustic and mechanical vibrations for resolving finer details on the dna or obtaining atomic resolution on the mica substrate .nevertheless , the measured dna height (  nm ) is in excellent agreement with earlier low - force tapping - mode experiments with much softer ( 1 n / m ) cantilevers , though not as close to the 2 nm expected from the diameter of ( uncompressed ) double - stranded dna as measured by tracking the resonance frequency of nanoscale cantilevers oscillating at subnanometer amplitudes .in summary , we have prepared cantilevers with attached beads , that show quality factors in water , yielding clearly recognizable resonances for different methods of cantilever actuation . this facilitates the application of dynamic - mode afm in liquids and provides an alternative method to overcome the influence of noise in deflection detectors used for afm .it will also facilitate the application of frequency - modulation afm in liquid : though the frequency - modulation technique is sufficiently robust against slight detuning of the phase , its operation becomes highly impractical when the frequency feedback loop can make discrete jumps between various spurious resonances close to the low- cantilever resonance ( see fig .[ fig : comparison]a , acoustic actuation ) . in spite of the orders - of - magnitude increase of the inertial mass of the cantilever , the ultimately achievable force sensitivity is hardly affected by the attachment of the bead to the cantilever .optimum results are obtained for beads that do not significantly exceed the cantilever width , such that the viscous damping remains dominated by the cantilever shape and not by the attached bead . the cantilevers with attached beadscan be well described by a simple mass - on - a - spring , damped harmonic - oscillator model , in which the effective mass is close to the bead mass .in addition , we have demonstrated and extensively documented the attachment of beads and the implementation of a magnetic actuation system using a standard afm combined with an inverted optical microscope .this will be of significant help for researchers interested in implementing these methods on any comined afm / inverted - optical - microscope system .we expect the results of this work to have broad applications for imaging soft , biological samples under near - physiological environments , and for accurate force measurements using dynamic - mode afm in liquids .we gratefully acknowledge adam mckay for the dna fragments , and magnequench for a donation of the magnetic powder .this work was funded by the uk biotechnology and biological sciences ( bbsrc , bb / g011729/1 and bb / j006254/1 ) and engineering and physical sciences ( epsrc , ep / k502959/1 ) research councils , and the us office of naval research ( n00014 - 10 - 1 - 0096 ) .( color online . ) design ( a ) and photograph ( b ) of the solenoid used for magnetic actuation .( c ) the voltage ( data points ) over the solenoid as a function of frequency , as measured using a standard digital oscilloscope while holding the ac drive voltage constant at about 2 v. the solid line represents a fit to the data using eq .[ eq : solenoid_voltage ] . ]  the solenoid for magnetic actuation was designed to fit into one of the objective threads of an inverted optical microscope ( ix71 , olympus , tokio , japan ) .it can thus easily be adopted for any afm system that is mounted on an inverted microscope platform . a sketch and a photograph of the solenoidcan be seen in fig .[ fig : coil]a - b .it consists of a cylindrical ferrite rod ( length = 30 mm , diameter = 4 mm , relative magnetic permeability = 31 ) , 42 turns of copper wire ( conductor diameter = 0.2 mm , insulation material : polyimide , insulation thickness = 0.08 mm ) and is fixed in a peek holder .it is connected to the external drive of the afm with a bnc cable . in our experiments ,a digital oscilloscope was used to monitor the voltage over the solenoid .the number of turns was optimized to generate a magnetic field in the frequency range of 10 khz , corresponding to the range of cantilever resonances in this work. a larger number of turns would increase the magnetic field for lower frequencies , but for higher frequencies would increase the solenoid impedance to such values that the current ( and thus the field ) becomes limited by the maximum voltage ( about  v ) of the excitation output of the afm controller .this behavior was modeled by considering the equivalent electric circuit including the drive voltage ( ) , the resistance of the excitation output and cables connecting the coil ( ) , the ohmic resistance ( ) of the coil and its impendance due to its inductance .the voltage over the solenoid ( ) then follows from with a fixed ( as measured independently ) , a fit to the measured ( fig .[ fig : coil]c ) yields and  h , which is in agreement with the prediction  h from the expression where is the cross section of the solenoid , the length , the relative permeability , and the number of turns of the solenoid . assuming a current and defining the central axis of the solenoid as the -axis , and the end of the solenoid at which the field is determined ( ) , the field along the axis is given by as derived by summing the fields of the individual current loops in the windings of the solenoid . for , this results in for the measurement range shown here, we did not find significant differences between the exact and the approximate equations .for magnetic actuation , the oscillation amplitude depends linearly on the magnetic field .[ fig : bofz ] shows the distance dependence of the oscillation amplitude in our setup .it demonstrates that the field decays relatively slowly on the length scale corresponding to practical solenoid - cantilever distances in an afm system .oscillation amplitudes of  nm were routinely achieved for cantilevers with attached magnetic beads .distance dependence of the magnetic field generated by the solenoid , as measured via the amplitude of the magnetically actuated cantilever ( dots , uncalibrated measurement ) .the cantilever with magnetic bead was immersed into a water filled petri dish such that the distance from the solenoid was a few mm .the solid line represents a fit with eq .[ eq : field ] , by setting  mm ( based on the core diameter and the thickness of the wires ) and  mm ( as measured ) , and allowing a variable offset in ( which defines the origin here ) .]  43ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1'''' [ 0]secondoftwosanitize [ 0 ] + 12$12 & 12#1212_12%12[1][0] link:\doibase 10.1016/s0167 - 5729(02)00077 - 8[ * * ,  ( ) ] link:\doibase 10.1002/wnan.104 [ * * ,  ( ) ] link:\doibase  10.1063/1.1866229 [ * * ,  ( ) ] link:\doibase 10.1063/1.2188867 [ * * ,  ( ) ] link:\doibase 10.1063/1.1999856 [ * * ,  ( ) ] link:\doibase  10.1063/1.2202638 [ * * ,  ( ) ] http://stacks.iop.org/0957-4484/20/i=26/a=264018 [ * * , ( ) ] * * ,  ( ) link:\doibase 10.1063/1.127051 [ * * , ( ) ] link:\doibase 10.1063/1.2753104 [ * * ,  ( ) ] link:\doibase 10.1063/1.2196052 [ * * ,  ( ) ] link:\doibase 10.1063/1.3053369 [ * * ,  ( ) ] link:\doibase 10.1063/1.3238484 [ * * ,  ( ) ] link:\doibase 10.1063/1.3689815 [ * * ,  ( ) ] link:\doibase 10.1063/1.117835 [ * * ,  ( ) ] http://dx.doi.org/10.1007/s003390100647 [ * * ,  ( ) ] * * ,  ( ) * * ,  ( ) link:\doibase  10.1143/apex.3.065205 [ * * ,  ( ) ] link:\doibase  10.1063/1.2804074 [ * * ,  ( ) ] link:\doibase 10.1063/1.2754076 [ * * ,  ( ) ] http://stacks.iop.org/0957-4484/19/i=38/a=384019 [ * * , ( ) ] link:\doibase  10.1063/1.1145130 [ * * ,  ( ) ] link:\doibase 10.1063/1.2200874 [ * * , ( ) ] link:\doibase 10.1016/j.sna.2006.11.034 [ * * ,  ( ) ] link:\doibase  10.1103/physrevb.80.134104 [ * * ,  ( ) ] link:\doibase 10.1063/1.371894 [ * * ,  ( ) ] link:\doibase 10.1063/1.2767202 [ * * , ( ) ] link:\doibase 10.1063/1.2794426 [ * * ,  ( ) ] link:\doibase 10.1063/1.3080557 [ * * ,  ( ) ] link:\doibase 10.1063/1.3658820 [ * * ,  ( ) ] link:\doibase 10.1063/1.3237180 [ * * ,  ( ) ] link:\doibase 10.1063/1.2654274 [ * * ,  ( ) ] link:\doibase 10.1063/1.347347 [ * * ,  ( ) ] link:\doibase 10.1063/1.3120913 [ * * ,  ( ) ] link:\doibase 10.1103/physrevlett.92.235501 [ * * ,  ( ) ] link:\doibase 10.1063/1.4745781 [ * * ,  ( ) ] link:\doibase 10.1021/nl301857p [ * * ,  ( ) ] link:\doibase 10.1016/s0304 - 3991(03)00004 - 4 [ * * ,  ( ) ] link:\doibase 10.1103/physrevb.74.195424 [ * * ,  ( ) ] link:\doibase 10.1063/1.3458009 [ * * , ( ) ] http://www.wiso - net.de / webcgi?start = a60&dokv_db=&dokv_no=9783868940817629&d% okv_hs=0&pp=1 [ _ _ ] ( , , )

Abstract: dynamic - mode atomic force microscopy ( afm ) in liquid remains complicated due to the strong viscous damping of the cantilever resonance . here we show that a high - quality resonance ( ) can be achieved in aqueous solution by attaching a microgram - bead at the end of the nanogram - cantilever . the resulting increase in cantilever mass causes the resonance frequency to drop significantly . however , the force sensitivity as expressed via the minimum detectable force gradient is hardly affected , because of the enhanced quality factor . via the enhancement of the quality factor , the attached bead also reduces the relative importance of noise in the deflection detector . it can thus yield an improved signal - to - noise ratio when this detector noise is significant . we describe and analyze these effects for a set - up which includes magnetic actuation of the cantilevers and which can be easily implemented in any afm system that is compatible with an inverted optical microscope .


the study of extreme events is a fundamental chapter in statistics and a pillar in its applications to natural phenomena . in recent years, this theory has been approached from a dynamical perspective . rather than arising from a stochastic process ,extreme events occur along the _ deterministic _ evolution of a system : see the review and references therein .this is not surprising , for chaotic properties of dynamical systems render them equivalent to stochastic processes , in a precise technical sense .this intuition guided the founders of modern non linear dynamics like joseph ford , to whose memory we dedicate this paper , and is still extremely fruitful .in this theory , one has been able to recover the three basic attraction laws of the statistical theory and to reproduce a wealth of its characteristics .this is highly desirable , since it is clear that natural phenomena follow deterministic , albeit chaotic , evolution laws . in this endeavor , the `` conventional '' approach let us call it so , for clarity has been to focus on the events which occur when the system s evolution approaches a _single _ point , or at most a finite collection of points , in a low dimensional phase space : the closer the approach , the more extreme the associated phenomenon . as a consequence , the function that gauges the strength of extreme events ( we shall call this function the _ intensity _ in what follows ) whether a distance , or a `` physical '' function , is rather simple .  in this paperwe want to pursue a different approach : we try to construct abstract models that , albeit amenable of analysis , reproduce the expected complexity of a _ physical landscape _ of the intensity function .consider in fact a realistic dynamical system modeling the evolution of a natural phenomenon .typically , its motions evolve in a many dimensional phase space , in which the intensity of a physical phenomenon is a non convex function with many local maxima , almost degenerate , or even singularities .we may think of projecting this evolution on a manifold of reduced dimension via any of the well known techniques : surfaces of section , embedding , _et cetera_. clearly , the graph of the intensity function so projected becomes a very complicated object , possibly displaying a fractal structure .alternatively , consider the minimization of a non convex , multi variable function a hard classical problem of computational mathematics . in algorithm has been proposed , by which a chaotic motion explores phase space , being attracted by minima when they appear in its neighborhood .this technique has been applied to inverse fractal problems and to model tuning for oil well production data .motions where close encounters with a scattered set of points have explosive significance may also evolve in real spaces : think of a robot `` sniffing '' the air in search of odor sources while cruising around .these sources can be mines in a field .probability of sensing a mine decays exponentially with distance ; mine distribution can not be assumed uniform and certainly / unfortunately more than one mine is to be expected .a further example is obtained by substituting the robot with a human and mines with infected individuals : spread of infections can be modeled this way , so that extreme values of a distance function become directly related to the probability of contagion .the above are just a few examples that motivate the investigation presented in this paper : we shall study dynamical extrema in low dimensional systems , when the intensity function has a complex , hierarchical structure .in so doing , while being closer to physics , we also introduce a new framework for mathematical investigation that reveals the _ dynamical _ rle of objects of deep significance , like minkowski dimension and content .  in summary, our work extends the conventional theory in some respects : first , as mentioned , rather than defining the intensity as the distance from a finite sets of points , we let it be a rather complicated function that we also derive from the distance from uncountable cantor sets .second , we consider the interplay of such sets and functions with motions with singular continuous invariant measure .this leads us to define generalized minkowski dimensions and contents . finally , we find that , under certain circumstances , limit laws hold also for merely ergodic dynamical systems with no decay of correlations .  previous related investigations with significant links to our results must be quoted , while underlying differences . in ,extreme value statistics for intensity functions taking their maximum value at a specific point of a fractal attractor have been studied .it has been shown that these statistics are linked to local dimensions of invariant measure on the attractor and can be used to reveal these quantities .singular continuous invariant measures have been considered in .these works , as well as other previous investigations we are aware of , are concerned with _ local _ quantities . to the contrary , we pursue here a _global _ approach .this difference emerges clearly when considering the relation between extreme value laws and hitting / return times statistics : conventional local quantities are replaced in our theory by global hitting / return times statistics , like those studied in . finally , still in the local framework ,extreme value laws have been shown to hold also in non mixing systems , via the peak over threshold approach , while in this paper they are observed within the block maxima technique .the approach adopted in this paper can be properly defined as experimental mathematics : while the definitions of the objects under examination , as well as the form of the conjectured limit laws , are complete and rigorous and , frequently , the value of various constants involved are obtained from explicit formal computations , we do _ not _ provide proofs of the conjectured results .we only put forward a heuristic argument in section [ sec - toy ] , which might perhaps lead to rigorous analysis .in fact , to the best of our knowledge , the examples presented herein belong to classes not studied so far that might require the development of new techniques .we therefore bring our contribution by exposing this new framework and by providing numerical evidence .let us now outline the organization of this paper .in the next section we sketch the conventional steps of dynamical extreme value theory , mainly to fix notations , and we describe the set - up of our numerical experiments .  in the third section we introduce a simple geometrical model of intensity function with a hierarchical structure : the cantor ladder not to be confused with the more common cantor staircase .readers who want to skip this specific example may go directly to section [ sec - discantor ] .we show that extreme value laws hold for this function , both for a chaotic dynamics and for a non - chaotic , just ergodic , dynamical system , with no decay of correlations .even if the form of the extreme distributions can be found _ exactly _ , it can not so far be proven _ rigorously _ , but only justified by numerical experiments .it therefore constitutes an interesting example for further investigation .the same remark also applies to the cases that follow .  in section [ sec - discantor ], we show that the previous example can be related to a generalization of the conventional setting described above , in which the intensity of a phenomenon is a function of the distance from prescribed points in phase space .in fact , we let these points populate a compact set , the ternary cantor set , and we show that the results obtained in the toy model of section [ sec - toy ] still hold .a particular case is then put in evidence : when the invariant measure of the dynamical system is the lebesgue measure in and the intensity function is the logarithm of the inverse distance to , extreme value statistics crucially depends on two quantities of fractal analysis : the minkowski dimension and the minkowski content of the set .this is a remarkable situation in which these quantities take on a dynamical meaning .  in section [ sec - genmink ]we introduce a further element : in the same setting of section [ sec - discantor ] , we consider dynamical systems with a generic invariant measure . in the conventional case ,such measures have been studied in , but still in relation to single point distance functions . in the proposed new framework ,extreme value laws depend on _ generalized _ minkowski dimension and content of the set , _ defined with respect to _ the measure .although quite natural , we have not been able so far to locate in the literature previous mention of these generalized quantities . when they exist , the results of the previous section are reproduced in this generalization .  to the contrary , in section [ sec - extrare ], we study a family of sets , a dynamical map of mbius type and an invariant measure ( also due to minkowski , the remarkable minkowski question mark function ) such that the conventional scaling law ( which yields non trivial dimension and content ) does not hold .a different law is derived and new minkowski constants are introduced , that reflect the non analytic behavior of the minkowski question mark function at rational values .physically , we interpret this law as describing _ extremely rare _ events . again , the new extreme value laws can be exactly computed and tested numerically with success .a particularly interesting instance of this non analytic behavior is studied in section [ sec - usuenne ] : this is the case of a set whose hausdorff and minkowski dimension differ .we derive the form of the scaling law of extreme events and we test it numerically . finally , the conclusion recap the highlights of this work in three paragraphs .in this section we briefly introduce the dynamical approach to extreme value theory , following the comprehensive review , and we describe the set up of our numerical experiments .consider a canonical dynamical system , comprising a transformation on a measure space , which preserves a probability measure which is also ergodic .also defined on is a physical observable , that represents the _ intensity _ of the phenomenon under examination .dynamics and observable permits to construct a stationary process , defined as for each .we let be the cumulative distribution function of the process : and we also let .consider now the evolution starting from , over a finite time span of length , and the maximum value reached by the intensity during this time interval .this defines the block maximum : an _ extreme value law _ for exists if there is a non - degenerate distribution function  ] and the cantor ladder function depicted in figure [ disegno ] , which can be compared to the more common cantor staircase function .this function takes constant values on every gap of the well known ternary cantor set , of increasing magnitude as the size of the gap gets smaller .this generates a prototype _ fractal landscape _ that models the situation described in the introduction .  to define formally this function one can use the technique of iterated function systems but in the present case we find it easier to proceed directly : write any number  ] composed of all points such that the first one in their representation is found at the -th position : for instance , , . clearly , these sets are the gaps of integer order in the usual construction of the cantor set , obtained by deleting middle thirds .equivalently , the set is the set of middle third intervals discarded at level in the ternary cantor set construction . therefore , is a collection of open intervals , of length and cardinality . for $ ] the discrete function  keeps track of the set to which it belongs .clearly , for lebesgue almost all the order is finite , we set it to infinity in the residual set .we stipulate that the physical function depends only on : and it is therefore a piece - wise constant function , whose value is constant on every set ( see fig .[ disegno ] ) . the monotonically increasing function of the integer argument will be specified shortly .whatever this choice , the function is discrete and is characterized by a countable infinity of _ plateaus _ on the gaps of a cantor set . on the cantor set itself , we take and to be infinity .  it is now time to specify the dynamics .we examine two different maps , both of which leave the lebesgue measure invariant .the first is the hyperbolic dynamics generated by the asymmetric tent map : , for , and for , with the parameter in .the second is the irrational rotation , with .as it is well known , this dynamics does not exhibit any decay of correlations .  on this basis , let us carry on the analysis sketched in section [ sec - extr ] . letting and , we easily find that the value is the self similarity ratio of the ternary cantor set .notice that the function is discontinuous and we are in the same situation discussed in : we adopt here the subsequence approach , to consider a discrete set of values and ( see below ) . also notice that is a label of the different level sets of and therefore of the intensity associated to the phase - space point . before making explicit reference to the function ,let us derive the underlying extreme value law associated with this dynamical process , assuming heuristically independence of the dynamical variables , which can be justified under the strong mixing properties of the dynamical system .we now write eq .( [ eq-2 ] ) as where is a function of : .in order that , as in eq .( [ eq-2 ] ) , we need that _ i.e. _ let us henceforth drop the term in and define the inverse function that yields the discrete sequence of values that can be realized , since the variable is discrete .then , we expect an extreme value law given by eq .( [ eq-3bis ] ) , with replaced by .the threshold values depend on the specific choice of the function .yet , the existence of an extreme value law depends on the discrete nature of this process and can be verified prior to such choice .in fact , let and , that is , we consider the extreme value of the integer function along a trajectory .then , it is clear that and eq .( [ eq-3bis ] ) , with , is equivalent to   this dependence can be verified by plotting versus and the combined variable : in figure [ fig1c.ps ] we observe convergence to the extreme value law , as grows . observe that _ three _ sets of data are reported in the figure and appear as almost undistinguishable see the successive figure [ fig3b.ps ] to appreciate differences .the first is the function ( that appears as a pure negative exponential , with the choice of axes of the figure ) ; the second is the plot of the numerical data for the dynamical system described above ; the third is analogous to the second , but under the dynamics .it is remarkable that also in this merely ergodic case the extreme value law is obtained in the limit .also observe that the extremal index takes the value one , in both cases ._ in lieu _ of a formal justification of these observations , which we defer to future works , we conjecture that the rle of fast decay of correlations in assuring the dependence conditions is played here by the complexity of the intensity function , a fact that could also explain the absence of clustering of extreme events .these results are better appreciated in figure [ fig3b.ps ] , where we plot the difference for the case of the map , which is larger than that observed for the case .as grows tends to zero for all values of . in this paper , we are not concerned with the rate of convergence to the limit distribution that could nonetheless be derived from these data . in the conventional framework ,convergence rates have been studied in .  finally , in figure [ fig4c.ps ] we also plot side by side the empirical densities for the tent map and the irrational rotation .the differences between the two fade away as grows and as they both tend to the limit extreme value law .we can put the above results in the usual casting of extreme laws by letting and thus re - writing eq .( [ eq - ca42 ] ) as \leq y \ } ) \rightarrow e^{-e^{-y}},\ ] ] where the variable is restricted to take discrete values , multiples of .we can now return to the problem of specifying the physical function . as typical in dynamical extreme value theory, we can choose different forms of the function .let us follow the standard classification given _e.g. _ in ref . , eqs .( 3.3 ) - ( 3.5 ) ; for reasons that will become clear in the following section , functions corresponding to the examples given in can be obtained by writing and making the substitution .this yields   case 1 has been just studied : in the other cases it is clear that the validity of the extreme value law for the underlying discrete process entails that of the derived processes . by straightforward calculation , since in eq .( [ eq - ca42 ] ) , we obtain that , letting in case and in case , the extreme value law \leq y\ } ) \rightarrow e^{-\theta \tau(j;y)}\ ] ] holds with that orderly correspond to the gumbel , frchet and weibull forms , respectively .let us now take a moment to comment on the second case . observe that this construction accounts for the gutenberg richter law , according to which the statistical frequency of events more intense than a threshold is a power law in .in fact , by choosing we obtain , which implies that ergodicity of the dynamical systems with maps and and lebesgue measure clearly implies that the phase - space average above is equal to a time average .the exponent is experimentally measured in a wide set of natural phenomena ( earthquakes , avalanches , floods , _ et cetera _ ) .we can so conclude that the simple model based on the cantor ladder well describes events with gutenberg richter statistics and yields a frechet law for extrema parameterized by the gutenberg richter exponent .relation between the two laws is well known in the natural sciences : for an example in geophysics , see .in the previous section we have described a simple dynamical model for extreme values that differs from standard usage in the literature in a significant respect : the intensity function is _ not _ related to the distance of the motion from a point ( or to the measure of a ball centered at ) , but it is a global function with a hierarchical structure . in this section we show that such model can be seen as an extension of the standard approach .we first show the analogy and we then develop the new theory .consider again the _ gap order function _ , that yields . recall that if and only if belongs to , _ i.e. _ to a gap of order in the hierarchical construction of the cantor set .since the cantor sets that we are considering are of null lebesgue measure , almost all points of a typical trajectory belong to a gap .then , the closest point of the cantor set to is one of the two gap extrema .define now the distance of a point to a cantor set as where compactness of permits to substitute the required with a minimum .the previous consideration permits us to compute the average value , with respect to the lebesgue invariant measure , of the _ logarithm _ of the distance when belongs to a gap of length / lebesgue measure : where . in our simple construction ,when , the diameter of the gap is and this proves that is proportional to the logarithm of the average distance of a point in a gap , from the cantor set .let us introduce the following generalization of the standard setting : rather than defining via the distance of from a finite set of points , replace the latter by a generic compact set .the fundamental definition now becomes in which is any of the three forms in eqs .( [ eq - trecas1a])([eq - trecas1c ] ) . in force of this analogy ,we _ conjecture that the extreme value laws obtained in the previous section for the cantor ladder model should correspond orderly to the choices of in eq .( [ eq - cant4 ] ) , when is the ternary cantor set . _  without loss of generality , let us henceforth concentrate on the case .the main object in the theory is the set where exceeds a threshold : in the new setting , this becomes the _neighborhood _ of radius of the set , a well known object in fractal geometry .clearly , in the case this reduces to the frequently investigated ball around the point , indicated by . taking the measure of the sets in eq .( [ eq - cant5 ] ) we obtain let us now consider again the case when the invariant measure in the above equation is the lebesgue measure and let the _ minkowski dimension _ of a set is defined as the difference , where is the euclidean dimension of the ambient space .it may , or it may not , coincide with the hausdorff dimension : it does , for the ternary cantor set ; we will see an instance of the opposite case in section [ sec - usuenne ] .upper and lower dimensions are defined in the standard way when the limit in eq .( [ eq - cant6 ] ) does not exist .furthermore , a set is defined to be _ minkowski measurable _ if the limit exists and is neither zero nor infinity .the constant is called the _minkowski content _ of the set .if we now follow standard derivations in dynamical extreme value theory we find that the _ local dimension _ at the point is now replaced by .recall that this is the case when the invariant measure of the dynamics , , is the lebesgue measure .we will generalize this situation in the following section .let us henceforth suppose that an asymptotic law holds , of the kind where is now the average minkowski content of the set ) exists only in a suitable cesro average , due to characteristic log - periodic oscillations .it is then called the average minkowski content . ] .then , and eq .( [ eq-2 ] ) implies that  these equations give the set of thresholds to be used in eq .( [ eq-3bis ] ) : we now test numerically the extreme value law .let be the ternary cantor set previously considered . in figure [ fig5b.ps ] we plot , in double logarithmic scale , the function versus , together with the asymptotic estimate of eq .( [ eq - cant61 ] ) .two remarks are in order at this point .the first is that parameters in eq .( [ eq - cant61 ] ) can be computed exactly : and .the second is that _ log periodic _ oscillations in are super imposed to the scaling behavior ( and the same happens for the statistics ). nonetheless , these oscillations , albeit theoretically important , are of inferior practical significance than the leading behavior .  as in the previous section ,we can then plot versus and , to verify the asymptotic law in which convergence is again to be understood in leading order , with super imposed oscillations .this is done in figure [ fig6b.ps ] in the case of irrational rotations , .convergence to the asymptotic law is evident . as was to be expected from the heuristic argument of the previous section ,the extremal index is again one .finally , we can resort to normalizing sequences , to show that we are in the presence of a gumbel type ( suitably translated results hold for the cases and ) extreme value law : in which the rle of the minkowski dimension _ and _ of the asymptotic constant , the minkowski content , is evident .this is a remarkable case in which these geometrical quantities take on a dynamical meaning .so far , we have considered dynamical systems whose invariant measure is the lebesgue measure .it is now interesting to consider measures significantly different from the latter : in particular , those that are singular continuous with respect to it . for sake of simplicity and with no loss of generality , we still consider the case of the previous section , when . as a consequence , recalling eq .( [ eq - cant5 ] ) , the function may now depend on the measure in a non trivial fashion .  to begin with , suppose that a relation like eq .( [ eq - cant61 ] ) holds , where and are now functions of the relation between the set and the measure .we therefore define the _ generalized minkowski dimension _ as the difference between the euclidean dimension of the embedding space and .likewise , the constant defines the _ generalized minkowski content _ , as in eq .( [ eq - dmin1c ] ) . under these circumstances , eq .( [ eq - cant62 ] ) still holds , with the obvious replacement of symbols , and we may expect that the previous conclusions on the extreme events statistics , eqs .( [ eq - cant62z])([eq - cant7 ] ) , also apply .let us now introduce an important dynamical system , defined by the map : this map is a sort of non - linear version of the map , this time composed of mbius transformations .it has a singular continuous invariant measure that appears in many problems of number theoretical origin . in dynamical systems, it has been studied for instance in , where a graph of its integrated density , can be found .the function is also known as the minkowski s question mark function .it satisfies the relations ,   \end{array}\ ] ] that equivalently serve to define it via a mbius iterated function system . among its remarkable properties one notices the fact that the derivative of exists and is null almost everywhere , but never on a full interval .in addition , has a non analytical behavior at all rational points , as we will see below .  in figure [ figc34f15 ]we plot the measure of the neighborhood of the ternary cantor set versus , and the asymptotic law ( [ eq - dmin1 ] ) , in which and . that is to say , for this measure of high theoretical importance , the ternary cantor set has minkowski dimension ( as in the lebesgue case ) but average minkowski content one !let us now examine the dynamical motions induced by the map and compute the extreme value statistics .we obtain the data reported in fig .[ figc34 - 13 ] where , at difference with the other figures , we have employed a linear scale for and a logarithmic scale for .convergence to the extreme value law is observed over many orders of magnitude , until fluctuations due to finite sampling emerge ( 6,400,000 samples of the variable have been computed for each value of ) .the results of this section can be explained heuristically by saying that the motion of the dynamical system samples space around the cantor set almost like the lebesgue measure , the difference emerging in the minkowski content while not in the minkowski dimension .this is certainly an important observation , that adds to the long list of non trivial behaviors of this dynamical system and calls for rigorous study .extreme value laws are linked to the statistics of rare events . in fact , consider for instance the dynamics studied in , where a selected point is chosen on the hnon attractor , at which the intensity of the simulated phenomenon is maximal .rare events occur when the motion enters balls around this location .typically , the measure of the ball scales in a power law fashion with its radius and the local dimension of the invariant measure exists as a positive number .we have extended this set up to neighborhoods of a set , via the generalized minkowski dimension and content . when these latter exist , results like those of the previous sections are to be expected .  yet, there are cases where the interplay between the exploring dynamical system and the compact set of singularities of the intensity function ( embedded , or not , in its attractor ) is more intricate than what observed at the end of the previous section . in these cases , different scaling properties than the one given in eq .( [ eq - dmin1 ] ) may hold , implying important differences in the extreme value distribution . in this sectionwe describe one of these cases that can be interpreted as describing the statistics of _ extremely rare events _ : in fact , in eq .( [ eq - expoball ] ) not only the radius of the set scales exponentially with the intensity of the phenomenon , but also its measure scales much more rapidly than power law in the radius .consider again the dynamical system defined in the previous section and the set . we know that , for small , , as can be shown using eq .( [ eq - mink2 ] ) . since , we are in the presence of a neighborhood of a set whose measure does not scale as eq .( [ eq - dmin1 ] ) .similarly , when , one computes . more generally , still using using eq .( [ eq - mink2 ] ) , it is possible to derive an asymptotic formula for the measure of all balls centered at points of the kind ( valid for and small ) : we are therefore in the presence of a dynamical system and a family of sets , the singletons , for which an asymptotic behavior different from eq .( [ eq - dmin1 ] ) holds : precisely , we would like to call the positive parameters , and the _ non standard , generalized minkowski constants_. of course , they no more possess the properties of a dimension . in eq .( [ eq - mink3 ] ) one has : the wider generality will be needed in the next section .consequences to the extreme value statistics are then important : therefore and .\ ] ]  observe the double exponential in the dependence of , or the double logarithmic dependence of on .they imply a sharp decay of the distribution function of the extremal variable , whose typical value increases with the logarithm of the logarithm of the time interval . in most realistic applications , this increase could be practically undetectable : it is a consequence of the _ extremely rare event _ .we can make use of the power of numerical experiments ( as opposed to real experiments ) to verify the previous conclusions : let us take and let us choose . as before ,validity of eq .( [ eq-3bis ] ) can be verified by plotting the experimental extreme value distribution versus and : this is done in figure [ figp25 ] , with theoretical parameters , and .the extremal index is one and the limiting curve ( blue ) is a pure exponential . in the same figure, we join by green lines the values corresponding to the same value of .difference between the experimental data and the theoretical limit curve are plotted in fig .[ figp25a ] .  observe that the functional dependence in eq .( [ eq - mink43 ] ) renders difficult to bring eq .( [ eq-3bis ] ) in one of the three standard forms and we are forced to parameterize the threshold as a function of . otherwise , allowing for a non linear scaling , we can obtain a gumbel law for the _ exponential _ of the extreme values : an exception to our approach , in the previous section we have considered a family of intensity functions with singularities concentrated at a single point . in this section we return to wider generality , by considering a set of infinite cardinality , compact , although not cantor , which provides an interesting family of extremely rare events .this set is a renown geometrical example : as it is well known , its hausdorff dimension is null ( it is a countable set of points ) while its minkowski dimension is one half , as it can be easily shown .let us first compute its generalized minkowski constants , when the measure is the derivative of the minkowski question mark function .we have to compute the of the of the set .observe that can be written as \cup \bigcup_{k = k_\varepsilon}^\infty   b_\varepsilon(\frac{1}{k}),\ ] ] where , .we can now use the results of the previous section to compute the measure of : setting , we obtain consider now the summation at r.h.s . :}.\ ] ] its behavior can be evaluated by a saddle point technique , with the result that , where and are minkowski constants ( that we do not attempt to compute explicitly here ) , so that observe that we have two different non standard behaviors appearing in this equation , both of which with a non unit exponent ( as different from the previous section ). evidently , the second term is leading , when tends to zero : we are therefore in the conditions of eq .( [ eq - mink4 ] ) , with the explicit determination , that constitutes still another generalization brought along by the theory .we can now perform the usual analysis , employing the the scaling induced by eq .( [ eq - mink41 ] ) .results are shown in figure [ figp25b ] .convergence to the asymptotic extremal law is again verified , but we observe numerically the presence of a non trivial extremal exponent .we leave the rigorous derivation of this exponent to future work , but we remark that its origin must be traced back to the fact that , since , the set is a trajectory of the motion .figure [ figp25c ] plots as before differences of experimental data and asymptotic theory .in this paper we have introduced a new paradigm in the dynamical theory of extreme events : the rle of fractal intensity functions , lacking any convexity or maximality property .we started by describing a toy model , the cantor ladder , that was then extended to general intensity functions with singularities on fractal subsets of the dynamical attractor .we have shown that the conventional theory can be generalized to these cases .the related extreme value laws bring into play familiar fractal quantities , the minkowski dimension and content , as well as they generalizations , defined herein , to singular continuous dynamical measures .remarkably , these laws appear to hold also for ergodic , non mixing systems .we have finally introduced the notion of extremely rare event and the corresponding extreme value laws , exemplified by a dynamical system with origin in number theory .we hope that the theoretical framework and numerical results presented in this paper will stimulate further analysis to prove rigorously the facts demonstrated by compelling numerical evidence , to clarify how complexity of the fractal landscape of the intensity function plays a rle in guaranteeing the validity of extreme value laws and at the same time be useful for applications of the theory to real phenomena .g. m. acknowledges hospitality at the isaac newton institute for mathematical sciences ( cambridge , uk ) during the 2013 programme _mathematics for the fluid earth _ , where this work has been started .generous support has also been provided by anr perturbations , anr valet and amidex for participation of gm to the workshop _dynamiques porquerolles 2015_. fruitful conversations with s. vaienti and j. freitas and the other participants to the two programs , whose names are too many to list , are also kindly acknowledged .computations for this paper were performed at the infn computer center at pisa .s. mantica , a. cominelli and g. mantica , _ combining global and local optimization techniques for automatic history - matching production and seismic data _ , society of petrolium engineers , _ spe elibrary _ paper number 78353 ( 2002 ) .

Abstract: typically , in the dynamical theory of extremal events , the function that gauges the intensity of a phenomenon is assumed to be convex and maximal , or singular , at a single , or at most a finite collection of points in phase space . in this paper we generalize this situation to fractal landscapes , _ i.e. _ intensity functions characterized by an uncountable set of singularities , located on a cantor set . this reveals the dynamical rle of classical quantities like the minkowski dimension and content , whose definition we extend to account for singular continuous invariant measures . we also introduce the concept of extremely rare event , quantified by non  standard minkowski constants and we study its consequences to extreme value statistics . limit laws are derived from formal calculations and are verified by numerical experiments .  _ keywords : extreme value laws , fractal landscapes , minkowski dimension , minkowski content , minkowski question mark function , mbius iterated function systems , extremely rare events , non standard minkowski behavior _ +  _ dedicated to the memory of joseph ford , on the twentieth anniversary of his departure _


as an established branch of mathematics , game theory analyzes strategic decision making of competing agents who find themselves in conflict situations . game theory finds extensive applications in social sciences , biology and engineering .recent developments in quantum computing and quantum information theory have motivated efforts to recast classical game theory using quantum probability amplitudes peres , leading to the birth of the area of quantum games mermin , mermin1,meyerdavid , ewl , vaidman , brandt , benjaminhayden1,benjaminhayden2,enkpike , johnson , johnson1,marinattoweber , iqbaltoor1,duli , du , piotrowski , piotrowski1,flitneyabbott1,iqbal , flitneyabbott2,cheontsutsui , shimamura , ichikawatsutsui , iqbalcheonm , iqbalweigert , ozdemir , iqbalcheon2007,flitneyhollenberg1,iqbalcheonabbott , ichikawa , iqbalabbott2008,chappell , flitneyrecent .quantum games have been proposed as a new way to approach quantum algorithms , including speculation that nature may be playing quantum games at the molecular level .  in the area of quantum games , a recently reported probabilistic approach constructs them from a set of non - factorizable joint probabilities .it is a set that can not be factorized in terms of other more fundamental probabilities .the motivating idea being that bell s inequalities can be violated by a set of non - factorizable probabilities , though this does not imply that any non - factorizable set will violate the inequalities . as the violation of bell s inequalities is considered a fundamentally quantum aspect , oneis motivated to have an approach to quantum games that constructs them from the property of a probability set of being non - factorizable .this then provides greater mathematical generality for quantum games , providing accessibility without the formalism of quantum mechanics .the earliest quantization attempts were focussed at the two - player two - strategy ( ) non - cooperative games .the playing of a game can be easily realized if players share a physical system that involves joint probabilities .this , for instance , will be the case when players share two coins that can be put in head ( ) and tail ( ) states .a referee gives each player a coin and asks them to flip it to either the or the state .as no tossing but only flipping actions are involved , it does not matter if the coins are biased or not .after a players flipping ( or not - flipping ) actions the coins can be found in a , , , or state , where the first entry in each pair , is reserved for the state of alice s coin . from a given game table , the referee can then award players their payoffs depending on the state of the two coins .the probabilistic approach to quantum games developed in ref .iqbalcheon2007 extends the playing of a game towards the quantum domain by considering two players who share a bigger physical system that also involves coin tossing .consider four biased coins that two players share to play a game according to the following arrangement . in a run , each player is given two coins and she / he has to select one .the referee tosses the two selected coins together and records the outcome .it can then be shown that the players payoffs , in a mixed - strategy version of the game , can be expressed in terms of their strategic choices performed over multiple runs and the relevant factorizable joint probabilities .  in order to maintain the bilinear payoff structure of the game ,constraints are placed on joint probabilities . allowing joint probabilities to become non - factorizable , while they remain under these constraints , ensures that the classical payoffs and the outcome of the game are obtained when the probabilities become factorizable .  in the present paper, we introduce an extra condition requiring that the considered joint probabilities are also symmetric both when they are factorizable and non - factorizable .we argue that this is a natural constraint to be included in a probabilistic approach to quantum games that builds them from non - factorizable joint probabilities .we find that this constraint further narrows down our focus in obtaining a quantum game from probabilistic considerations only .we study quantum versions of the prisoners dilemma ( pd ) game , the stag hunt ( sh ) game , and the chicken game within this approach that constructs them from not only non - factorizable but also symmetric joint probabilities .we investigate how non - factorizable and symmetric joint probabilities permit new equilibria in these games . by introducing parameters that present a measure of non - factorizability , we discuss a novel way of obtaining a set of non - factorizable joint probabilities , which allows us to identify factorizable , non - factorizable , and the quantum domains .we begin with the table of sixteen joint probabilities as shown in table raweprprobabilities .  for alice and bob we also have payoff matrices given by alice s and bob s payoffs , respectively . because we are considering games with symmetrical payoffs we have , where indicates transpose .this requires    in a mixed - strategy game one has the strategy vectors and , with  ] is the probability with which alice selects coins and  ] are defined to be the probability to select over and over by alice and bob respectively , then we have alice s expected payoff given by s payoff is then obtained from above by interchanging and . substituting eqs .( [ strategypayoffs ] ) and re - arranging we find  , and with  and . by symmetrywe also have for bob a ne we need to satisfy the relations \geq 0  \notag \\ \pi _ { b}(x^{\ast } , y^{\ast } ) -\pi _ { b}(x^{\ast } , y ) & = & ( y^{\ast } -y)[x^{\ast } \delta _ { 3}v_{3}+\delta _ { 1}v_{1}-\delta _ { 2}v_{2}]\geq 0 .\label{symmetricneconditions}\end{aligned}\]]we thus have the ne defined for symmetric games with the three variables .if the probability table is factorizable then we can write  ] or or create the classical mixed ne in the classical game , we define where  ] .this gives us or or that .this result indicates that and are a proportional distance from .this then gives that the other ne , , if we have then we require from eq .( [ alicefactne ] ) \geq 0,\]]and in order to return requires .also , if we have then we require \geq 0,\]]and in order to return requires .hence we find three ne on eq .( [ rcondition ] ) and eq .( [ scalc ] ) .we know , hence we can always find an and an to create this particular classical game .for the chicken game we have and and , where . the general condition for neis obtained from eq .( [ alicefactne ] ) as \geq 0.\]]we can see that hence we will duplicate the results of the previous sh game , obtaining the correct classical ne  and .we find , hence for the three games studied , if we select such that with given by eq .( scalc ) , in each case we will return the classical ne for these three games when the probability table becomes factorizable , although not the classical bilinear payoffs . if we also require this payoff structure , then we require the more restrictive constraint and .we have shown that factorizability along with symmetry and the conditions and embeds the classical game within the quantum game .if we enforce in the general quantum game , then we have by normalization , but by symmetry , and therefore , similarly for the rest of the table .hence this condition creates the table of factorizable probabilities with and all other probabilities zero .however we can still create a non - factorizable set of probabilities by inserting offset parameters from the starting position in the table ( factorizedtable ) .we now add extra parameters into table factorizedtable exploiting any available degrees of freedom .for the upper left quadrant because we are constrained by normalization , symmetry , and the causal communication constraint , we only have available two degrees of freedom .this is utilized with the parameters and as shown in table ( [ factorizednonfact ] ) .we then continue this process and we find that we can add up to independent parameters , in the range  ] .  from table ( [ factorizednonfact ] ) and eqs .( [ vdefn ] ) we find we can parameterize the ne in terms of the three parameters , and . substituting into eq .( [ symmetricneconditions ] ) we find \notag \\ & = & ( x^{\ast } -x)(r - s)[\delta _ { 3}\{y^{\ast } ( r+\frac{\epsilon _ { 1}-\epsilon _ { 3}}{r - s})+(1-y^{\ast } ) ( s+\frac{\epsilon _ { 1}}{r - s})\}-\delta _ { 2}(1+\frac{\epsilon _ { 1}-\epsilon _ { 2}}{r - s } ) ] .\notag \\ & &  \label{nonfactorizablenealice}\end{aligned}\]]the corresponding inequality for bob is then obtained by interchanging and .this gives us the general conditions for a ne in the non - factorizable case .note that if we set we recover our previous results in eq .( alicefactne ) and eq .( [ bobfactne ] ) .it is also easily shown that this condition also implies that and so we will recover the factorizable payoff relation shown in eq .( [ alicepayofffactorizable ] )   the embedding of the classical game is obtained by taking and , which from eq .( [ nonfactorizablenealice ] ) gives the equation for ne  \notag \\ & & \pi _{ b}(x^{\ast } , y^{\ast } ) -\pi _ { b}(x^{\ast } , y )  \notag \\ & = & ( y^{\ast } -y)[\delta _ { 3}\{x^{\ast } ( 1-\epsilon _ { 3})+\epsilon _ { 1}\}-\delta _ { 2}(1+\epsilon _ { 1}-\epsilon _ { 2 } ) ] .\label{nonfactorizableclassicalembedding}\end{aligned}\]]thus we have obtained the general conditions for the ne for a non - factorizable table of probabilities , which will embed the classical game when it becomes factorizable . the payoff given by +y[(a_{3}-a_{4})(1-\epsilon _ { 2})+\epsilon _ { 1}(a_{1}-a_{2 } ) ]  \label{nonfactorizableneclassicalembeddingpayoff}\end{aligned}\]]and similarly for bob .  these produce table [ factorizednonfactclassicalembedding ] and we see that we must have .   cereceda finds the chsh sum of correlations for any set of local hidden variables satisfying the causal communication constraint as table [ factorizednonfactclassicalembedding ] , we can find the table of probabilities , we note that  ] exist in agreement with the expected range . for example , using , we find .however , quantum mechanics enforces extra restrictions on the joint probabilities considered here that can arise , namely cirelson s bound cirelson of  ] ,  ] , so that any mixed ne we desire in the range $ ] , as well as returning to the classical ne when . if we desire to produce the non - classical ne of and , then from eq .( nonfactorizableclassicalembedding ) we have the conditions have for the sh game and so we can see that if we select , then we will have achieved this new ne .this condition gives or .this is easily satisfied , with with the other terms zero , for example .the chicken game is defined with whereas the sh game has .thus we can carry over the results from the previous section , except that the ne will invert due to the extra minus sign in eq .( [ nonfactorizableneclassicalembedding ] ) .quantum versions of games are developed considering the peculiarities of a set of quantum mechanical joint probabilities .the probability sets we consider consist of normalized probabilities satisfying causal communication and the symmetry constraints .players are allowed classical strategies only and their payoff relations are re - expressed in terms of the joint probabilities .we allow a quantum game thus defined to reduce itself to the classical mixed - strategy game when the set of joint probabilities can be factorized in terms of the factorization parameters and .constraints on the parameters and are obtained with which this reduction can be realized .non - factorizable sets of joint probabilities are introduced and appropriate parameters and describing non - factorizability are identified .quantum games are now constructed by retaining the obtained constraints on the parameters and and allowing non - factorizability parameters and to take non - zero values .two types of games are identified : firstly , with , and , , we obtain the original classical mixed - strategy game along with its bilinear payoff structure . secondly , while enforcing and , but allowing non - factorizability parameters to take non - zero values , we obtain an extension of the classical mixed - strategy game in which the full original classical game , along with its bilinear payoff structure , remains embedded when .we investigate pd within this setup to find that when cirelson s bound is maximally saturated , a non - factorizable and quantum game gives the ne of at which both players payoffs approach to their pareto optimum value . for the shwe observe that two new and non - classical ne of  can be realized with non - factorizable joint probabilities .we demonstrate that the our non - factorizable extension of the classical game permits us to study situations that are not even physically realizable .that is , the situations in which the corresponding chsh inequality is violated beyond cirelson s bound .we then obtain a constraint , given by eq .( quantumgamerestriction ) , that defines the boundaries of what quantum mechanics can permit for the extension of the original classical game .notice that the new parameters and are introduced in order to give an idea of the extent of how much non - factorizable a given table of joint probabilities is relative to the factorizable situation .so as to obtain compact expression , the parameters are then introduced , each of which depends on and .these extra parameters are added numerically to the and parameters _ without _ affecting the meaning of the and parameters .it turns out that and subsequently become redundant in our quantum game as we set and in order to embed the classical game within the quantum game .the extension of game theory based on the considerations of quantum mechanical joint probabilities attaining the peculiar character of being non - factorizable has already been investigated in earlier publications iqbalcheon2007,iqbalcheonm , iqbalcheonabbott , iqbalabbott2008 .the present manuscript s contribution consists in understanding how placing an extra symmetry requirement on joint probabilities changes the role of non - factorizability in the construction of quantum games . using probabilistic considerations only , this work explores quantum games that are constructed using an epr type setting .this approach gives a more accessible perspective on the nature of quantum mechanical joint probabilities and their potential exploitation in giving an extension to game theory .one important benefit of this extension is the extended perspective it provides of looking at the quantum mechanical probabilities that is able to cover the classical factorizable , non - factorizable , and even those situations that quantum mechanics does not allow , within a single framework .quantum mechanics is a probabilistic theory and this paper holds that probabilistic considerations permit us to have a more clear vision and sense of what quantum mechanics can achieve and what are its limits . from this viewpointwe give an extension to game theory , while focussing on purely probabilistic considerations .we observe that this extension is general enough to show us the classical factorizable situations as well as the situations that are beyond quantum mechanics .we are motivated to have an entirely probabilistic approach towards quantum games that encompasses classical , quantum and also those hypothetical situations that can not be realized quantum mechanically .we allow players the same sets of classical strategies so that this scheme is not subjected to enk and pike type argumentation . as an epr type apparatusis used in the playing of the two - player quantum games , their physical realization will involve performing epr type experiments . these experiments are agreed to entail genuinely quantum features .the sets of joint probabilities , whose non - factorizable property we use in constructing our quantum games , are relevant to generalized epr type experiments .the potential benefits of this approach consists of developing an entirely probabilistic understanding of multi - party strategic situations . in these situations , quantum probabilities become crucial in achieving one or the other outcome .the extension of game theory advocated in this paper , rather than being imaginary , is simply a more generic framework , that allows us to consider , in entirely probabilistic terms , also that peculiar domain which resides beyond the bounds of quantum mechanics .we find that , within our probabilistic approach toward quantum games , this domain becomes more easily identifiable , along with easily recognizable classical factorizable and the quantum mechanical non - factorizable domains .

Abstract: we construct quantum games from a table of non - factorizable joint probabilities , coupled with a symmetry constraint , requiring symmetrical payoffs between the players . we give the general result for a nash equilibrium and payoff relations for a game based on non - factorizable joint probabilities , which embeds the classical game . we study a quantum version of prisoners dilemma , stag hunt , and the chicken game constructed from a given table of non - factorizable joint probabilities to find new outcomes in these games . we show that this approach provides a general framework for both classical and quantum games without recourse to the formalism of quantum mechanics .