it is well known that the toplogy of the network changes the dynamics dramatically . after the discovery of the absence of epidemic threshold in a scale - free network, many researchers have focused on the dynamics on complex network , such as synchronization , pattern formation , and other phenomena .  in this paper , we study the bouchaud - mzard ( bm ) model on complex network .the power - law behavior of wealth distribution , pareto s distribution , has for over a century been one of the main cornerstones of econophysics .one of the simplest models that explain this law is that proposed by bouchaud and mzard , which consists of multiplicative noise and globally coupled diffusion . after the proposal of the bm model , several researchers numerically investigated the generalized bm model in which diffusion occurs between adjacent nodes in a complex network .while , the research revealed that the network topology alters the wealth distribution , there has been no quantitative theory that sufficiently explains these simulation results .recently , we proposed a new theory for the bm model on a random network .using several assumptions that we describe in the next section , we derived equations that determine the static probability density function ( pdf ) of the wealth .the results of this analysis were compared with those of the numerical simulations and good agreement was obtained .however , the wealth distributions were analyzed only on a random network in the previous paper , and the distributions on other complex networks was left as an open problem .the aim of this paper is develop our previous work so that it is applicable to a general complex network .using the same techniques applied in the previous paper , we derive the equations that determine the static pdf for the bm model on a complex network for a given adjacency matrix .the results are evaluated by a comparison with the numerical simulation results , and our method is revealed to perform well for many network systems .this paper is organized as follows . in section[ theory ] , we derive the self - consistent equations that determine the pdf of the bm model on a complex network whose adjacency matrix is given .the results obtained with this pdf are tested in sec .[ simulation ] by comparing with simulations on several networks such as a random network , the watts strogatz ( ws ) network , and a real social network .finally , we discuss the results and future problems and summarize the paper in sec . iv .we consider the bm model on a undirected complex network consisting of nodes , whose adjacency matrix is given by .the dynamics of , i.e. , the wealth on node , are determined by the following ito - type stochastic differential equations . where , and represents the diffusion constant of wealth , strength of noise , and the standard brownian motion , respectively .this equation has no static pdf ; however , the normalized wealth , where represents an average over all nodes , can have a static pdf .for simplicity , is used in place of in the following discussion .  to obtain a static distribution, we make the `` adiabatic and independent '' approximation introduced in our previous paper .we first assume that all values are independent and that the correlation between different nodes is negligible . under this assumption ,we assume the pdf can be decomposed as .second , we assume that the rate of change of , the average around node , is much slower than that of , where represents the degree of node . using these assumptions , we can calculate the static pdf as follows .suppose is constant .then , the conditional pdf of on node for a given , is a solution of the following fokker planck equation . + \sigma^2 \frac{\partial^2}{\partial x^2 }  [ x^2 \rho_i]\;.\ ] ]  the static solution of this equation , , is given by where and .using the adiabatic assumption , the static pdf of at node is given by where represents the pdf of .  to obtain , we use the independent assumption again together with the central - limit theorem . under this assumption, can be assumed to be the distribution of the average of the independent variables , where represents a node adjacent to node .if the variances of are finite for all , we can apply the central - limit theorem to approximate by the gaussian distribution \;,\label{091257_8aug12}\ ] ] where and represent the average and variance of , respectively . from lindeberg s theorem ,these values are given by and where and are the average and variance of , respectively .we can determine and by calculating and for using eqs .( [ 164143_7aug12 ] ) , ( [ 164204_7aug12 ] ) , ( [ 090008_8aug12 ] ) , and ( [ 090021_8aug12 ] ) . by using and for , we find and where we assume .we thus obtain for all , and is the solution of using eq .( [ 090021_8aug12 ] ) , we can rewrite this equation as for convenience and in anticipation for later analysis , we define a new matrix as we then finally obtain the following equation for : eqs .( [ 164143_7aug12 ] ) , ( [ 164204_7aug12 ] ) , ( [ 091257_8aug12 ] ) , and ( [ eqfors ] ) define the pdf of .  before concluding this section , we note in the following about wealth condensation , defined as the divergence of . because for all , eq . ( [ eqfors ] ) must have a solution to have any meaningful result .if is large enough , this condition is satisfied provided that . for a value that satisfies for all , is a non - negative matrix .therefore , the solution of eq .( [ eqfors ] ) satisfies if , where represents the spectral radius of . from frobenius s theorem , is equal to the largest eigenvalue of , and thus we conclude that eq .( [ eqfors ] ) has a meaningful solution if the largest eigenvalue of is less than 1 .noting that , , and for , is less than 1 for large , and a non - condensed phase appears , provided that .in contrast , we always have a condensed phase at small due to the divergence of the elements .as we decrease from a large value , the values increase and diverge when ; also diverges in this case .therefore , the wealth is condensed if .to test our theory , we compare the results with numerical simulation results on several networks .we begin with the network created by the erds - rnyi algorithm and then consider several networks created by the ws model to test the effect of clustering .it is shown that our theory fails when the rewiring rate is too small , but good agreement is obtained when . finally , we test our theory on a real network. it would be ideal to apply the theory to a real economic network , but as there is currently no available data , we use the american college football network obtained by girvan and newman instead . for this case , the pdf obtained from the simulations shows good agreement with our theory .for all the simulations , is set to 1 .the network created by the erds - rnyi algorithm , shown in fig .[ 140327_24aug12 ] , has 200 nodes and a mean degree of .  in fig .[ 170117_22aug12 ] we plot the pdfs obtained by numerical simulation for , and 2.0 .the pdfs obtained using our theory are indicated by the solid line and show good agreement with the numerical simulation for all for the case , but there are minor discrepancies at small values in the cases of and 0.5 .the discrepancy is due to the assumption used for deriving eqs .( [ 170529_22aug12 ] ) and ( [ 170556_22aug12 ] ) . the maximum value calculated by eqs .( [ eqfors ] ) and ( [ 133408_24aug12 ] ) is 0.33 for and 0.09 for , while .thus , the approximation is not valid as is larger than 0.3 .however , the approximation is valid for the case as the maximum value is 0.01 .next , we consider three networks created by the ws algorithm .the networks have nodes and a mean degree of with different rewiring rates of , and 0.2 .[ 155309_15aug12 ] shows both the networks and pdfs . in the case of ,our theory differs from the numerical simulation , while we find good agreement for the and cases .+  to investigate the reason for this discrepancy , we examine the scatter plots of and for and in fig .[ scatterplot ] . for , we find that there is a strong correlation between and that weakens for .the differences can be quantified by the correlation , which is 0.74 for and 0.48 for .this result suggests that the discrepancy is caused by the breakdown of independent assumption .it is reasonable to question here when and why the independent assumption fails .the small - world property does not appear to affect the independence of wealth because the networks show high - clustering and small betweenness for both the and 0.1 cases .we return to this point in the discussion section .the american college football network obtained by girvan and newman includes 115 nodes and 613 edges .the simulated pdfs for and 0.5 are shown in fig .[ 093508_20aug12 ] . in both cases ,our theory reproduces the pdf well .the agreement between theory and simulation seems excellent .we have not understand the reason for this good agreement , however , one possible reason is that this network has community structure . as girvan and newman showed , this network has clear communities . as we will see in the later section ,the spatial correlation of wealth would be small for globally coupled network .therefore it is natural to assume that wealth correlation is also small , if the network is consisted from several densely connected subgraphs .however , we still need further study to explain this excellent agreement .here , we have developed a theory for the bm model on complex networks . by generalizing our previous work, we were able to propose a theory applicable to a complex network with a given adjacency matrix . using the adiabatic and independent assumptions, we derived the equations that determine the static pdf of the wealth .the result was compared to numerical simulations , and we found our theory works well on a erds - rnyi network , ws networks with , and the american college football network .we also found that the theory did not perform well for a ws network with as the independent assumption is no longer valid .the theory does not apply to a ws network with a small value because of the large spatial correlation in this case .thus , we may ask when and why the spatial correlation becomes large . while it is difficult to give a complete answer to this question, the following discussion suggests that the properties of the laplacian are essential to this problem .suppose that is very large and the fluctuations of are very small .under such circumstances , we can approximate in eq .( [ 083834_8aug12 ] ) as and obtain where is the laplacian matrix . to investigate the correlation in this toy model ,we assume for simplicity that the network is undirected and connected .introducing the eigenvalues and corresponding normalized eigenvectors of as and , we can write . then is diagonalized and eq .( [ 150804_20aug12 ] ) can be written as the static distribution of is calculated as \;.\label{133706_28aug12}\ ] ] the term does not have a static distribution because . however , because , this term gives the uniform change , , and does not contribute to the correlation .  from eq .( [ 133706_28aug12 ] ) and the relation , we obtain the covariance our independent assumption implies that the off - diagonal elements of eq .( [ 145452_24aug12 ] ) are smaller than the diagonal elements .this condition is satisfied if every is `` localized '' , in other words , has only one large components .it has been shown by some researchers that eigenvectors is localized in many complex networks .our theory will be applicable to these network models .we also consider here the case for which the adiabatic assumption is not valid .if the node degree is large enough , then changes at a much slower rate than that of .therefore , this assumption will be valid if the degree of each node is large .on the other hand , if the degree of each node is small , then not only the adiabatic assumption but also the use of the central - limit theorem can not be justified .for example , our theory will not work well for the `` star''-network in which almost all nodes have a degree of 1 .we note that theory for the bm model in the wealth - condensed phase is still elusive , as the standard central - limit theory is not applicable when diverges .thus , to discuss the static properties , we will need to generalize the central - limit theorem so that it can be adopted even if the variance diverges .finally , we note that our developed technique is applicable to other dynamical systems subject to noise .stochastic dynamics on complex network is attracting the interests of many reserchers recently , however , dynamics with multiplicative noise has not studied so far .we believe this method will shed new light on other problems regarding the dynamics on complex networks .99 t. nishikawa , a. e. motter , y .- c lai , and f. c. hoppensteadt , phys .lett . * 91*,014101(2003 ) .t. ichinomiya , phys .e * 70 * 026116(2004 ) .h. nakao and a. mikhaikov , nature phys .* 6 * 544(2010 ) .j. bouchaud and m. mzard , physica a * 282 * 536 ( 2000 ) .v. pareto , _cours dconomie politique _ macmilan , london 1897 .d. garlaschelli and m. i. loffredo , physica a. * 338 * 113(2004 ) .w. souma , y. fujiwara , and a. aoyama , _ arxiv : cond - mat/0108482v1_. t. ichinomiya , arxiv:1209.2467v1 .d. j. watts and s. h. strogatz , nature * 393 * 440 ( 1998 ) . m. girvan and m. e. j. newman , proc .usa * 99 * 7821(2002 ) .p. n. mcgraw and m. menzinger , phys .e 77 , 031102 ( 2008 ) .h. nakao and a. mikhailov , nature physics * 6 * 544 ( 2010 ) .

Abstract: we study the wealth distribution of the bouchaud mzard ( bm ) model on complex networks . it has been known that this distribution depends on the topology of network by numerical simulations , however , no one have succeeded to explain it . using `` adiabatic '' and `` independent '' assumptions along with the central - limit theorem , we derive equations that determine the probability distribution function . the results are compared to those of simulations for various networks . we find good agreement between our theory and the simulations , except the case of watts  strogatz networks with a low rewiring rate , due to the breakdown of independent assumption .


measurements of the vector magnetic field in the solar atmosphere are crucial for understanding a wide variety of physical phenomena . using the linear polarisation of magnetically sensitive spectral lines to infer the component of the field transverse to the line - of - sight results in an ambiguity of 180 in its direction .this ambiguity must be resolved in order to completely determine the magnetic field vector .several methods are currently in use to resolve the ambiguity ( for overviews see ; ) . in general, each method involves an assumption or approximation that may not be appropriate for observations of solar magnetic fields .  a promising approach , which can potentially avoid unrealistic assumptions , is to exploit the divergence - free property of magnetic fields ( _ e.g. , _ ; ; ; ; ; ) .one challenge in calculating the divergence is that it requires information about the variation of the field in the direction perpendicular to the solar surface . to this endit is possible to derive from observations the variation of the field along the line - of - sight direction ( _ e.g. , _ ; ; ; ; ; , ; ; ; ; , ) .the line - of - sight direction is not perpendicular to the solar surface , except at disk centre ; therefore some care is required to correctly treat the geometrical perspective .in  ( , henceforth paper i ) we showed that the ambiguity can be resolved with line - of - sight and horizontal heliographic derivative information by using the divergence - free property without additional assumptions about the nature of the field . to resolve the ambiguity away from disk centre we demonstrated that it is sufficient to determine four pieces of information : _i ) _ the sign of the line - of - sight derivative of the line - of - sight component of the magnetic field ; _ ii ) _ the sign and magnitude of the line - of - sight derivative of the magnitude of the transverse component of the field ; _ iii ) _ the sign and magnitude of the line - of - sight derivative of the azimuthal angle ; and _ iv ) _ the sign and magnitude of the horizontal heliographic derivatives of the magnitude of the transverse component of the field and the azimuthal angle .paper i was a theoretical examination in which we assumed that all necessary derivatives could be determined exactly at a given location and that the solar surface could be represented by the heliographic plane .the aim of this paper is to investigate how the divergence - free property can be used to resolve the ambiguity for data where the magnetic field is measured at discrete spatial locations . using synthetic data we test two methods that have previously appeared in the literature ( _ e.g. ,_ ; ; ; ; ) ; both of these are based on the divergence but they make different assumptions about how to use it to resolve the ambiguity . we show that both of these methods can produce significant errors . for this reason , we present a more robust algorithm , the global minimisation method .the outline of this paper is as follows . in section [ sec_divb ]we describe the derivation of the divergence - free condition for any position on the solar disk in terms of observable quantities and discuss the consequences for ambiguity resolution . in section [ sec_synth ]we review the synthetic data and metrics that will be used to test the performance of the ambiguity resolution algorithms . in section [ algorithms ]we test the algorithms and the validity of the assumptions made . in section [ sec_conc ]we draw conclusions .within a limited field of view we assume that the solar surface can be represented by the heliographic plane , which is tangent to the solar surface at the position given by the central meridian angle , the latitude , and the - and -angles . over the heliographic plane ,the image components of the magnetic field are related to the heliographic components according to the following transformation :    where the coefficients are taken from equation ( 1 ) of , is the line - of - sight component of the field , and the components of the field perpendicular to the line - of - sight direction are given by    where is the magnitude of the component transverse to the line - of - sight and the azimuthal angle , , is the angle between the transverse component of the field and the -axis ( where and are coordinates on the plane perpendicular to the line - of - sight ) . the azimuthal angle can only be determined observationally within the range .the derivative in the direction perpendicular to the heliographic plane , , can not be directly measured away from disk centre . however , the derivative along the line - of - sight direction , , can be measured ( see below for further discussion ) .these derivatives are related as follows :    where is any differentiable function , is the distance along the line - of - sight direction , and and are coordinates on the heliographic plane .derivatives with respect to horizontal heliographic distance , and , can be approximated with techniques such as finite differences and finite elements using discrete measurements over the heliographic plane . for convenience ,we work with the line - of - sight distance , although in practice the field is likely to be inferred as a function of optical depth ; in which case a model atmosphere must be used to determine the correspondence between line - of - sight distance and optical depth ( _ e.g. , _ ; ; ; ) .equations ( [ b_h ] ) and ( [ dlos ] ) can be used to express the divergence of the field in terms of observable quantities ( _ i.e. , _ derivatives of the image components of the field with respect to , , and ) ,    where    equations ( [ divb2 ] ) and ( [ da ] ) show that the computation of the divergence requires the line - of - sight derivatives of all three components of the magnetic field vector , except at disk centre where the coefficients ( see paper i for further discussion ) .several techniques have appeared in the literature to measure the line - of - sight variation of the magnetic field ( often but not always including the full magnetic field vector ) .we briefly discuss three that could be employed to evaluate the various terms in equations ( [ divb2 ] ) and ( [ da ] ) .the first and possibly most straightforward is to measure the field at different heights using multiple spectral lines ( _ e.g. , _ ; ) .the inferred height difference between measurements can be several hundred kilometres or more depending on line choice , although exact heights can not be easily quantified due to the lines often broad range of formation height .a second approach makes use of a single ( generally chromospheric ) line that forms over a range of heights .inversions performed at multiple , but limited wavelength ranges near ( for example ) the core and separately in the line wing can return the field strengths at different atmospheric heights ( _ e.g. , _ ; ) .again , estimates of the field are obtained at discrete but broad heights which may be separated by tens or hundreds of kilometres .a third approach recovers the line - of - sight stratification of the magnetic field during the inversion of the stokes spectra through a formalism based on response functions ( _ e.g. , _ ; ; ; , ; ; , ) .the field is computed at discrete heights ( inversion nodes ) which may be separated by several hundred kilometres ( _ e.g. , _ ) ; interpolation can map the results onto a grid with points separated by as little as 5 kilometres . for all of the aforementioned techniques , the fieldis effectively sampled discretely .therefore , the line - of - sight derivatives in equations ( [ divb2 ] ) and ( [ da ] ) must be approximated .it is important to understand that the inferred line - of - sight derivatives of the transverse components of the field , and , depend on the ambiguity resolution at all heights used to approximate line - of - sight derivatives .consequently , for ambiguity resolution algorithms based on these derivatives , vector magnetogram data should be disambiguated simultaneously at all relevant heights .compared several techniques for disambiguating single - height magnetogram data for which information about the variation of the field out of the heliographic plane was not available . to test the performance of the various algorithms based on the divergence we use the same magnetic field configurations that were used by for which the correct ambiguity resolution is known and magnetic field data are available at two heights .methods based on the divergence require magnetic field data from at least two heights .all methods examined in this paper could be adapted for data with more than two heights ; however , this can add a degree of complexity to the disambiguation process ( depending on the algorithm ) because all heights may need to be disambiguated simultaneously .for each synthetic magnetogram , the field is sampled at discrete spatial locations .the grid points are intended to mimic pixel centres , although the exact value of the field is provided .thus , the effects of noise and unresolved structure in both the line - of - sight and horizontal heliographic directions are neglected ( see _ e.g. , _ ) ; these issues are beyond the scope of this investigation and will be considered in a separate paper .the grid points at each height are aligned in the line - of - sight direction and separated by a line - of - sight distance , .  ;positive / negative vertical magnetic flux is indicated by _red_/_blue _ contours at 100 , 1000 , 2000 g , and the magnetic neutral line is indicated by the _black _ contour .horizontal magnetic field is plotted at every fifth pixel , with magnitude proportional to arrow length .tickmarks are in units of pixels .( a ) vector magnetic field at from the numerical simulation of fan and gibson ( 2004 ) at their time step 56 ( is a length scale , effectively the size of the simulation domain in the -direction ) .the pixel size is .this is an exact reproduction of figure 1 from metcalf _ et al . _ ( 2006 ) .( b ) vector magnetic field for the multipole structure in image coordinates .the pixel size is .this is an exact reproduction of figure 2 from metcalf _ et al . _ ( 2006 ) ., title="fig:",scaledwidth=49.0% ] ; positive / negative vertical magnetic flux is indicated by _red_/_blue _ contours at 100 , 1000 , 2000 g , and the magnetic neutral line is indicated by the _black _ contour .horizontal magnetic field is plotted at every fifth pixel , with magnitude proportional to arrow length .tickmarks are in units of pixels .( a ) vector magnetic field at from the numerical simulation of fan and gibson ( 2004 ) at their time step 56 ( is a length scale , effectively the size of the simulation domain in the -direction ) .the pixel size is .this is an exact reproduction of figure 1 from metcalf _ et al . _ ( 2006 ) .( b ) vector magnetic field for the multipole structure in image coordinates .the pixel size is .this is an exact reproduction of figure 2 from metcalf _ et al . _ ( 2006 ) ., title="fig:",scaledwidth=49.0% ]  the field shown in figure [ synth](a ) is a snapshot from the numerical mhd simulation of of a highly twisted flux tube emerging into an overlying potential field `` arcade '' . because the simulation results are provided on a three - dimensional rectangular grid this case corresponds to disk centre , where the line - of - sight and -directions are parallel .figure [ synth](a ) shows the field at the first height ( as used by ) ; the second height used in this investigation lies above this , and is discussed in detail below .  the field displayed in figure [ synth](b )is constructed from a collection of point sources located on a plane below the surface .the contribution to the field from each source is calculated using the green s function given by with a constant value of the force - free parameter .this magnetogram is located away from disk centre , centred at n , w , and it includes the effects of curvature .again figure [ synth](b ) shows the field at the first height ( as used by ) ; the second height lies above this and is discussed further below .  to quantify the performance of the various algorithms we will use the same four metrics presented in : the fraction of pixels correctly disambiguated , the fraction of magnetic flux with the correct solution , the fraction of strong horizontal field regions (  g ) with the correct solution , and a measure of the discrepancy in the vertical current between the retrieved and correct solution , ; note that can have values outside the range $ ] unlike the other three metrics .thus the results presented here can be directly compared to those obtained for the algorithms examined by .in this section we test several disambiguation methods all based on the divergence - free condition but which employ different assumptions and implementation details .first , we examine two methods that have previously appeared in the literature : the criterion ( _ e.g. , _ ; ; ; ) and the sequential minimisation method ( _ e.g. , _ ) .we find that both of these methods can produce significant errors for both synthetic data sets .consequently , we present and test a third algorithm , the global minimisation method .   one non - iterative approach that was originally proposed by and tested on synthetic data by , , and , involves expressing the divergence - free condition as an inequality . for any position on the solar disk ( except the limb ) the divergence - free conditioncan be written as    assuming that the various derivatives can be determined reliably , given the sign of the ambiguity can in principle be resolved by choosing the azimuthal angle such that satisfies equation ( [ divb1 ] ) . if or then equation ( [ divb1 ] ) can not be used to resolve the ambiguity .consequently , for discrete observations equation ( [ divb1 ] ) can not be used if or if , where is an estimate of the measurement error in the respective derivatives .one advantage of using equation ( [ divb1 ] ) to resolve the ambiguity is that only the sign of is required , not its magnitude ; note , however , that the sign and magnitude of and are required away from disk centre ( see paper i ) .one limitation of using equation ( [ divb1 ] ) is that the inferred divergence of the field is not exactly zero ( in general ) when approximated from discrete observations , so the inequality is not exact ( ) .  ( a ) ) ._ contours _ are the same as figure [ synth](a ) , except that the _ green _ contour is the magnetic neutral line .areas with the correct / incorrect ambiguity resolution are _black_/_white _ ( the same convention as metcalf _ et al .. initial configuration has .( b ) same as ( a ) except the initial configuration of azimuthal angles is random.,title="fig:",scaledwidth=49.0% ] ( a ) ) ._ contours _ are the same as figure [ synth](a ) , except that the _ green _ contour is the magnetic neutral line .areas with the correct / incorrect ambiguity resolution are _black_/_white _ ( the same convention as metcalf _ et al ._ , 2006 ) .initial configuration has .( b ) same as ( a ) except the initial configuration of azimuthal angles is random.,title="fig:",scaledwidth=49.0% ]  the algorithm proceeds as follows . at a given height each column of pixels is scanned from to , and different columns are disambiguated from to .hereafter , the indices , , and correspond to pixel labelling in the - , - , and -directions , respectively , and and refer to the number of pixels in the - and -directions , respectively . for and at a given height we approximate the horizontal heliographic derivatives and at pixel with three - point finite differences using the measurements at pixels : , , and . at disk centre , where the heliographic and image coordinates are equivalent ,these approximations reduce to standard two - point forward finite differences . at the boundaries and the geometry of the finite differencing stencil is modified to use only pixels within the field of view . at the lower / upper height forward /backward finite differences are used to approximate the line - of - sight derivative using measurements from the two heights ..performance metrics for the flux tube and arcade at disk centre for the various ambiguity resolution algorithms ( with  pixel ) .the lower height is evaluated . [ cols="^,^,^,^,^",options="header " , ]   the assumption made by this method is tested by using the correct solution to evaluate each of the terms in equation ( [ divb1 ] ) with the various derivatives approximated as described above . in figures [ inequality ] and [ inequality_tpd7 ] areas that satisfy the inequality in equation ( [ divb1 ] ) for the correct solution are shown along with areas that violate equation ( [ divb1 ] ) . also shown are areas which suffer from the degenerate case where equation ( [ divb1 ] ) can not be used to distinguish the two realisations , as the sign of is the same for both the correct and the incorrect choice of azimuthal angle at the pixel of interest .this is the case for a large fraction of the field of view for both synthetic data sets .the method appears to do quite well in figure [ fan_ts56](a ) and figure [ compare_tpd7_wuai](a ) because we do not test the other realisation if the initial one satisfies equation ( [ divb1 ] ) and we do not check that equation ( [ divb1 ] ) is satisfied after the choice of azimuthal angle is changed .the degeneracy is partly due to the details of the implementation and could be avoided by changing the azimuthal angle in all pixels referenced by the finite differencing stencil at a particular location , rather than at a single pixel ; unfortunately , such an approach would be very sensitive to the initial configuration ._ contours _ are the same as figure [ fan_ts56 ] .areas that satisfy / violate the wu and ai criterion , equation ( [ divb1 ] ) , are coloured _ black_/_white_. areas for which the sign of is the same for both choices of the azimuthal angle are coloured _ gray_.,scaledwidth=49.0% ]  ( b ) ) ; _ black_/_white_/_gray _ indicate areas which satisfy , do not satisfy , and are degenerate to the inequality in equation ( [ divb1 ] ) , following figure [ inequality ] .( b ) same as ( a ) except for the second height ., title="fig:",scaledwidth=49.0% ] ( b ) ) ; _ black_/_white_/_gray _ indicate areas which satisfy , do not satisfy , and are degenerate to the inequality in equation ( [ divb1 ] ) , following figure [ inequality ] .( b ) same as ( a ) except for the second height ., title="fig:",scaledwidth=49.0% ]   presented a method which assumes that the magnitude of the divergence is minimised over the set of pixels used to approximate the divergence at a particular location .disambiguation then proceeds as a sequence of minimisation problems based at each pixel in the field of view .we apply this approach as follows . at a given height ,we examine pixels in the same order as in section [ sec_wuai ] . for the pixels referenced by the finite differencing stencil at a particular location we choose the permutation of azimuthal angles that corresponds to the minimum of , the approximation for the divergence at the pixel .we compute the divergence using the methods described in section [ sec_wuai ] , with all approximations made about the lower height .the azimuthal angle at each pixel is fixed by the stencil that first references it .this approach examines both permutations of the azimuthal angle at each pixel , therefore the initial configuration does not affect its performance , although it is sensitive to the order in which pixels are examined .the performance of this method is shown for the flux tube and arcade ( figure [ compare_min_local](a ) ) . at disk centrethe disambiguation at each height is independent because and are not required to compute the divergence and the approximations for horizontal heliographic derivatives involve measurements at a single height ; results for the lower height are displayed in figure [ compare_min_local](a ) . for the multipole field positioned away from disk centre ( figures [ compare_tpd7_min_local](a ) and ( b ) ) both heights are disambiguated simultaneously as required for off - disk - centre data .performance metrics for the lower height in each case are provided in tables [ fan_tab ] and [ tpd7_tab ] .  .( b ) validity of the assumption made by the sequential minimisation method .areas that satisfy / violate the assumption are coloured _black_/_white_. _ contours _ are the same as ( a ) . , title="fig:",scaledwidth=49.0% ] .( b ) validity of the assumption made by the sequential minimisation method .areas that satisfy / violate the assumption are coloured _black_/_white_. _ contours _ are the same as ( a ) . ,title="fig:",scaledwidth=49.0% ]  ( a ) .( b ) same as ( a ) except for the second height , 1 pixel ( or ) above that shown in ( a ) .( c ) validity of the assumption made by the sequential minimisation method applied to the multipole field , following figure [ compare_min_local](b ) .( d ) same as ( c ) except for the second height ., title="fig:",scaledwidth=49.0% ] ( a ) .( b ) same as ( a ) except for the second height , 1 pixel ( or ) above that shown in ( a ) .( c ) validity of the assumption made by the sequential minimisation method applied to the multipole field , following figure [ compare_min_local](b ) .( d ) same as ( c ) except for the second height ., title="fig:",scaledwidth=49.0% ]  ( a ) .( b ) same as ( a ) except for the second height , 1 pixel ( or ) above that shown in ( a ) .( c ) validity of the assumption made by the sequential minimisation method applied to the multipole field , following figure [ compare_min_local](b ) .( d ) same as ( c ) except for the second height ., title="fig:",scaledwidth=49.0% ] ( a ) .( b ) same as ( a ) except for the second height , 1 pixel ( or ) above that shown in ( a ) .( c ) validity of the assumption made by the sequential minimisation method applied to the multipole field , following figure [ compare_min_local](b ) .( d ) same as ( c ) except for the second height ., title="fig:",scaledwidth=49.0% ]  for each pixel the ambiguity is resolved by the stencil that references it first .hence , for the pixels in a given stencil this algorithm assumes that the correct ambiguity resolution will be recovered for the unresolved pixels if the previously resolved pixels have the correct ambiguity resolution .we show areas for which this assumption is valid for both test cases in figure [ compare_min_local](b ) and figures [ compare_tpd7_min_local](c ) and ( d ) .evidently , this assumption is not universally valid when the divergence is approximated from discrete observations .however , the fraction of pixels that violate this assumption is smaller than the fraction that retrieve the incorrect azimuthal angle .there are two reasons that this method fails at a particular pixel : _i ) _ the assumption is incorrect ; and _ ii ) _ the propagation of an erroneous solution that originates in a region where the assumption is incorrect .propagation accounts for most of the areas with the incorrect azimuthal angle in figure [ compare_min_local](a ) and figures [ compare_tpd7_min_local](a ) and ( b ) .most , but not all , pixels that violate the assumption initiate the propagation of errors . also , there are pixels that violate the assumption but still retrieve the correct ambiguity resolution ; these situations can occur when the assumption is violated and a neighbouring pixel has the incorrect ambiguity resolution , allowing the algorithm to get back onto the correct solution .the algorithm described above is different from the implementation of in that they used four - point finite elements to approximate horizontal heliographic derivatives ( we note that there may also be other differences ) .we have tested other versions of this algorithm employing different approximations for the horizontal heliographic derivatives , such as four - point finite differences and finite elements .the results are very similar to those shown in figure [ compare_min_local ] and [ compare_tpd7_min_local ] : all versions examined violate the underlying assumption at some locations and subsequently propagate errors in the solution .this method can propagate errors because it is sequential .we have tried a non - sequential approach which assumes that is minimised within the stencil based at each pixel regardless of the solution obtained in neighbouring pixels . for both synthetic datasets the fraction of pixels for which this assumption is violated is actually larger than for the sequential approach .therefore , we conclude that the non - sequential approach does not substantially improve the performance of this method .in this section we present an iterative , global method which is based on the approximation for the divergence summed over the entire field of view .this approach is related to the `` minimum energy '' algorithm for single - height data ( see _ e.g. , _ ; ; ; ) .we assume that the correct configuration of azimuthal angles over the field of view is the one that corresponds to the minimum of    equation ( [ esb ] ) consists of two terms corresponding to the approximations for the divergence based at each of the two heights , and . for data positioned away from disk centre , where both heights are disambiguated simultaneously ,we find that algorithms based on equation ( [ esb ] ) consistently retrieve a better solution than algorithms based on a single term . for the divergence at the first / second heightthe line - of - sight derivatives are approximated with forward / backward finite differences .for each of these terms the horizontal heliographic derivatives are approximated using three - point finite differences as described in section [ sec_wuai ] .we have tested several different ways to approximate horizontal heliographic derivatives , such as four - point finite differences and three- and four - point finite elements , but these do not significantly affect the performance of the algorithm .the number of possible solutions for this optimisation problem is finite but very large , , making a brute - force search for the global minimum of impractical .one method that works well for large , discrete , global optimisation problems such as this is simulated annealing ( _ e.g. , _ ; ; ) . using simulated annealing , the global minimisation method proceeds as follows .a pixel is selected at random .if changing the azimuthal angle at that pixel reduces the value of then the change is always accepted .if changing the azimuthal angle increases the value of then the change is sometimes accepted , with a probability given by    where is the `` temperature '' and is the change in ( equation ( [ esb ] ) ) caused by changing the azimuthal angle ; positive values of imply that increases due to the reconfiguration .the ability to sometimes accept an `` uphill '' reconfiguration enables simulated annealing to avoid getting stuck in a sub - optimal solution , provided that the cooling rate is sufficiently slow and the initial temperature is sufficiently high ( _ e.g. , _ ) .  the annealing schedule ( the initial temperature , the number of reconfigurations attempted at each temperature setting , and the cooling rate ) can require some experimentation ; in the interests of simplicity we use the following prescription . the initial temperature , , should greatly exceed any expected value of . starting with the initial configuration of azimuthal angles we examine reconfigurations of randomly chosen pixels ( which are all accepted ) and take two times the maximum value of as the initial temperature .it should be noted that this process effectively randomises the initial configuration of the azimuthal angles and therefore the initial configuration does not affect the final result ( likewise , the high initial temperature further randomises the initial configuration because almost all reconfigurations will be accepted during early temperature increments ) . at each temperature setting we examine reconfigurations of the azimuthal angle at randomly chosen pixels .the temperature varies according to , where is the cooling rate and is the label of the temperature increment ; we use .we continue in this fashion until one of the following three conditions is satisfied : _i ) _ no reconfigurations are accepted at the most recent temperature setting ; _ ii ) _ the temperature is sufficiently low ( less than ) ; or _iii ) _ for ten consecutive temperature increments the value of does not change substantially ( _ i.e. , _ , where is the value of at the end of temperature increment ) .  for the flux tube and arcade positioned at disk centre, we show the fraction of pixels correctly disambiguated by the global minimisation method , , as a function of the vertical grid spacing ( figure [ dz_plot ] ) ; only the lower height is evaluated because the disambiguations at each height are independent at disk centre for the derivative approximations employed here .the simulation results of have a vertical grid spacing of ( equal to the horizontal grid spacing ) .for the first height we use the snapshot displayed in figure [ synth](a ) at .we allow the second height to vary , using all heights in the range . for each setting of disambiguate twenty times with the algorithm described above but with a different sequence of random numbers for each disambiguation .independent disambiguations can behave differently because simulated annealing is a stochastic optimisation algorithm : the decision to move to higher is probabilistic ( equation ( [ pup ] ) ) and the order in which pixels are examined is random .consequently , if the annealing schedule is not ideal ( _ e.g. , _ if the cooling rate is too fast ) the global minimum may not be found and a different answer may be obtained for a different sequence of random numbers . to ensure that the global minimum is found for real magnetogram data it may be necessary to repeat the disambiguation process multiple times with different sequences of random numbers , a slower cooling rate , and/or with more attempted reconfigurations at each temperature setting .  , as a function of vertical grid spacing ( in units of horizontal pixels ) for the global minimisation method with a fixed annealing schedule applied to the flux tube and arcade configuration ( figure [ synth](a ) ) . for each setting of disambiguations are executed with different sequences of random numbers .the _ gray _ dots represent the resultant value for each of these disambiguations .the _ black _ line connects the median of the values for each setting and the _ gray _ lines connect the corresponding minimum and maximum of the values .the _ dotted _ horizontal line at is included for reference.,scaledwidth=49.0% ]  for the flux tube and arcade at disk centre with  pixel the global minimisation method retrieves the correct azimuthal angle at every pixel for all twenty disambiguations with the imposed annealing schedule ( see table [ fan_tab ] ) .several points are worth noting regarding figure [ dz_plot ] : _ i ) _ broadly speaking , the performance decreases as increases . _ ii ) _ the scatter of the values increases as increases ( most noticeably for  pixels ) ._ iii ) _ for  pixels the correct solution is no longer found for all twenty disambiguations .the maximum of the values is one at this point so the correct solution could be obtained simply by disambiguating multiple times and selecting the solution with the smallest value of . _iv ) _ the point where the maximum of the values deviates from one is  pixels . beyond this point ,a perfectly correct solution is not obtained for any of the twenty disambiguations with the imposed annealing schedule , yet for the largest values of the median value for is still greater than 0.99 .pixels , following figure [ fan_ts56 ] .( a ) the solution retrieved with the best value .( b ) the solution retrieved with the worst value ., title="fig:",scaledwidth=49.0% ] pixels , following figure [ fan_ts56 ] .( a ) the solution retrieved with the best value .( b ) the solution retrieved with the worst value ., title="fig:",scaledwidth=49.0% ]  for large values of the difference between the best and worst solutions retrieved by the global minimisation method can be significant ( figure [ compare_fan_me ] ) .the main cause of such deviations is the annealing schedule .we have conducted experiments that show that more consistent and better results can be obtained at large values of with annealing schedules that involve slower cooling rates or more attempted reconfigurations at each temperature setting .  for the multipole field positioned away from disk centre the variation of as a function of the line - of - sight grid spacing is shown ( figure [ dz_plot2 ] ) ; both heights are disambiguated simultaneously as required for off - disk - centre data .for the first height we use the field displayed in figure [ synth](b ) .we allow the second height to vary in increments along the line - of - sight direction corresponding to the horizontal pixel size , .of the twenty disambiguations with  pixel , only two solutions obtained for both heights ( see table [ tpd7_tab ] ) ; the remainder of solutions obtained or better at each height .the performance of the global minimisation method applied to the multipole field configuration ( figure [ dz_plot2 ] ) degrades more rapidly as increases than it does for the flux tube and arcade ( figure [ dz_plot ] ) .this is partly because the optimisation problem for data positioned away from disk centre is substantially more challenging , as both heights must be disambiguated simultaneously .differences in the variation of the two fields with height may also contribute to the contrasting performance .the synthetic data for the multipole field configuration include the effects of curvature but we have not included curvature in the treatment of the divergence ( equations ( [ divb2 ] ) and ( [ da ] ) ) .experiments with two synthetic data sets that have the same field configuration but include and exclude curvature indicate that curvature does not significantly influence the performance of the global minimisation algorithm , provided that the field of view is much smaller than the radius of curvature of the sun .  in figure [ dz_plot2 ], we see that for some values of larger than 31 pixels ( _ red _ dots in figure [ dz_plot2 ] ) , the algorithm retrieves solutions with less than that for the correct solution .thus , the assumption that the correct solution corresponds to the configuration of azimuthal angles that minimises ( equation ( [ esb ] ) ) is violated in these cases . nevertheless , the performance of the resulting solutions is quite good , with values greater than 0.99 at both heights ( see _ e.g. , _figure [ compare_tpd7_2 ] ) . for larger than53 pixels no such solutions are found , which probably indicates that the fixed annealing schedule employed here is less than ideal for large values of ., as a function of vertical grid spacing ( in units of horizontal pixels ) for the global minimisation method with a fixed annealing schedule applied to the multipole field configuration in figure [ synth](b ) , following figure [ dz_plot ] .the _ red _ dots represent the values for each of the disambiguations that retrieved a solution with less than that for the correct solution .( b ) same as ( a ) except for the second height ., title="fig:",scaledwidth=49.0% ] , as a function of vertical grid spacing ( in units of horizontal pixels ) for the global minimisation method with a fixed annealing schedule applied to the multipole field configuration in figure [ synth](b ) , following figure [ dz_plot ] .the _ red _ dots represent the values for each of the disambiguations that retrieved a solution with less than that for the correct solution .( b ) same as ( a ) except for the second height ., title="fig:",scaledwidth=49.0% ]  results for a case where the retrieved solution has less than that for the correct solution are shown for the multipole configuration with  pixels ( figure [ compare_tpd7_2 ] ) .interestingly areas that fail occur at different locations at each height ; this situation is fairly typical for  pixels . for smaller values of incorrect regions tend to occur at similar locations at both heights .pixels , following figure [ compare_tpd7_wuai](a ) . for this particular case for the retrieved solution is less than that for the correct solution .( b ) same as ( a ) except for the second height , 50 pixels above that shown in ( a ) ., title="fig:",scaledwidth=49.0% ] pixels , following figure [ compare_tpd7_wuai](a ) . for this particular case for the retrieved solution is less than that for the correct solution .( b ) same as ( a ) except for the second height , 50 pixels above that shown in ( a ) ., title="fig:",scaledwidth=49.0% ]  for the flux tube and arcade we have not found a solution for which is less than that for the correct solution , in the case where the horizontal resolution is fixed but the vertical resolution varies ( figure [ dz_plot ] ) . we find that this situation does arise for the flux tube and arcade when both the horizontal and vertical sampling are about a factor of seven times coarser than on the original grid .clearly , the validity of the underlying assumption made by the global minimisation method that the correct configuration of azimuthal angles corresponds to the minimum value of  is sensitive to the spatial resolution of the observations .this suggests that for real magnetogram data ( that may contain noise and/or unresolved structure ) additional constraints or criteria may be required to determine if the global minimum corresponds to the correct solution .it may also be possible to use measurement error information to determine the uncertainties associated with the configuration corresponding to the global minimum .we investigate how the divergence - free property of magnetic fields can be exploited to resolve the azimuthal ambiguity that is present in solar vector magnetogram data by using line - of - sight and horizontal heliographic derivative information approximated from discrete measurements . using synthetic data we test several methods that each make different assumptions about how to use the divergence to resolve the ambiguity .we find that all of the assumptions considered can be violated when the divergence is approximated from discrete measurements ( depending on the nature of the field and the resolution of the observations ) , but some assumptions are more robust than others .the criterion expresses the divergence - free condition as an inequality ( _ e.g. , _ ; ; ; ) .we find that methods based on this criterion are very sensitive to the smoothness of the initial configuration of azimuthal angles and the order in which pixels are examined .in addition , the underlying assumption can be incorrect over a significant fraction of the field of view when derivatives are approximated from discrete measurements .the sequential minimisation method ( ) assumes that the magnitude of the divergence is minimised over the pixels used to approximate the divergence at each point .our tests indicate that this approach 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Abstract: we investigate how the divergence - free property of magnetic fields can be exploited to resolve the azimuthal ambiguity present in solar vector magnetogram data , by using line - of - sight and horizontal heliographic derivative information as approximated from discrete measurements . using synthetic data we test several methods that each make different assumptions about how the divergence - free property can be used to resolve the ambiguity . we find that the most robust algorithm involves the minimisation of the absolute value of the divergence summed over the entire field of view . away from disk centre this method requires the sign and magnitude of the line - of - sight derivatives of all three components of the magnetic field vector .


besiii is a modern spectrometer located at the upgraded beijing electron positron collider ( bepcii ) , which runs in the energy region ( 2 - 4.6 gev ) and aims at -charm physics .particle identification ( pid ) plays an essential role in the experimental study .one of the main sub - detectors , the time - of - flight ( tof ) system , responding for trigger and pid , consists of a barrel and two endcap .its capability of pid is determined by the flight time difference of particles species and the time resolution of the detector .the current tof system based on plastic scintillation and photmultiplier is built in 2001 .the calibration result of endcap tof system show that the time resolution for electrons in bhabha events is 148 ps , which is significantly worse than the resolution for mouns in dimu events ( 110ps , the designed goal ) .it s also found that scattering in mdc endplate materials can significantly influence the measured times of electrons in bhabha events .these findings indicate that the contribution to timing performance from multiple scattering interaction in the endcap material between the main drift chamber ( mdc ) and the etof is important .improvement is needed for the etof system to better meet the besiii physics goals .the r&d for this upgrade began in 2010 .monte - carlo ( mc ) simulation serves as an important tool to guide the upgrade , by comparing the performance of tof based on different technologies and optimizing the prototype design . in this paper a detailed simulation study , including all main features of the besiii etof system , is performed . the geant4 package ( geant version 4.09.02.p01 ) , commonly used in high energy experimental physics , is taken as the simulation tool .the besiii etof system is located between a helium - based multilayer mdc and a csi(tl ) crystal calorimeter ( emc ) inside the besiii spectrometer .the cross - sectional view of the etof system is shown in fig.[1 ] .please note although not shown in the figure , there are some materials , such as cables and readout electronics equipment , located between the endplate of the mdc endcap and the etof .    because the description of the detector structure and materials in the full besiii mc framework is quite complicated and difficult to modify and tune ,we develop a simplified model to do this job for the etof system .the main structure in our simulation is shown in fig.[2 ] .basically it consists of three parts .the left part is the mdc volume , filled with a gas mixture of 60% he and 40% c3h8 .the field and sensitive wires in the mdc are not simulated , since their contribution to multiple scattering is small .the middle part , including the endplate of mdc and the readout electronics and cables , contains the major material budget between the mdc and etof .the composition and equivalent thickness of the materials in this part are estimated by using the full besiii mc framework .virtual ( non - interacting ) particles are emitted toward the mdc endcap region with   ( is the polar angle as shown in fig.[2 ] ) .for each step the virtual particle travelled , the material composition and equivalent thickness ( along z direction ) are recorded and accumulated until the virtual particle hit the etof .we find four major media dominate the material budget , namely aluminum , printed circuit board ( pcb ) , copper and plastic .their equivalent thicknesses are 21.12 mm , 9.77 mm , 0.58 mm and 9.59 mm , respectively .the right part is the etof system , in which the detailed structure depends on the detector technology used and will be addressed in the following part of the paper .the magnetic field of besiii is a uniform field of 1 tesla along the z direction .the current tof endcap , each consisting of 48 trapezoidal - shaped plastic scintillation ( bicron 404 ) modules , is located at 1330 mm away from the interaction point ( ip ) along the beam direction ( z axis in the global besiii coordinate system , as in fig.[2 ] ) . for each scintillator module ,the length is 431 mm and the thickness is 48 mm , coving an azimuthal angle range of 7.5 , as shown in fig.[3](c ) .more details can be found in reference .to simulate the bhabha events , which play an important role in the offline calibration of besiii tof system , electrons with momentum of 1.5gev / c are emitted from the ip . the injection direction with respect to the z axisis chosen to be  so that the extrapolated hit point locates near the center of the etof module in the radial direction . in the simulationwe use standard geant4 electromagnetic ( em ) physics process including ionization , multiple scattering , bremsstrahlung and the gamma interaction with default settings for production thresholds of secondary particles .particles hitting the etof are electrons , positrons and gamma s .the hit position distributions on etof module , of primary electron , secondary electron / position and gamma , are shown in fig.[4](a),(b ) and ( c ) respectively .the etof module boundaries are also shown in the figures . in order to test the worst case ,the extrapolated hit points are chosen to be at the center of an etof module ( so the hit multiplicity is highest ) .    to further understand the feature of the secondary charged particles and the effect on etof timing , we show in fig.[5 ] the radial hit position distribution on etof of the secondary electron and positron , as a function of transverse momentum ( pt ) .two notable bands can be seen in the figure , one with higher pt and r close to 64 cm , while another with very low pt and r close to 54 cm .combined with fig.[4](b ) , we can conclude a rough scenario that the band near r=64 cm mainly comes from bremsstrahlung gamma conversion to electron / position pair , mostly following the direction of primary electron , while another band located at r=54 cm are those low energy ( thus also low pt ) electrons and positions generated from the em shower generated by the primary electron or from energetic ionization . in strong magnetic field these low energy electrons or positions basically travel along the field line , thus their hit position on etof reflects the position where they are generated , which is r54 cm from fig.[2 ] ( 1.1 m ( ) ) .    for each primary electron ,a large number of secondary electron , positron and gamma are produced in mdc endcap region causing multi - hit on a scintillator module .more quantitatively , we define the multi - hit rate as the fraction of events with 2 or more hits on one etof readout cell .we find the multi - hit rate of scintillator - based etof is about 71.5% .multi - hit can badly influence the timing performance of etof by distorting the output signal shape and amplitude that are hard to be calibrated at offline .note there is no tracking information available for these secondary particles in the calibration .  to study the effect of the complex hit position distribution structure and the high multi - hit rate on etof timing , we make a semi - quantitative calculation by evaluating the hit time and position information .for each track , the measured arrival time is the sum of particle flight time from the ip all the way to etof and the signal s transmission time to the readout end .we assume a simple one - dimensional linear timing dependence on the hit position . in the scintillator, the signal transmission time can be expressed as ( r - r0)/v , where r0 is the radial position of the photomultiplier , and v is the effective transmission velocity in scintillator .r0 is 47.2 cm in current besiii etof design and v-1 is measured to be 80 ps / cm . in each event ,the earliest arrival time at the photomultiplier among all hits is taken as the measured tof .the tof distribution is shown in fig.[6 ] . beside the nominal tof peak at around 6.2 - 6.4ns caused by primary electrons , another peak with much smaller tof around 5.5 ns is clearly visible in fig.[6 ] .the left peak is about 0.9 ns earlier in time than the nominal peak - this is understood as a consequence of secondary electron / positron band at low energy and low r in fig.[5 ] . in fig.[5 ] , the difference in r for the two main bands is 10 cm .consider the transmission velocity of fluorescent photons in scintillator , their arrival time difference will be 0.8 ns , agreeing well with the fig.[6 ] .the difference in flight time from the ip to etof is relatively small , since electron or position travels at a velocity very near to the speed of light .our simulation also confirms this point .furthermore , similar feature was observed with the full besiii mc framework ( including fluorescent photon generation and transmission ) , as well as the calibration results with experimental bhabha data collected by besiii .the consistency of these results validates the reliability of our simulation .the simulation results of scintillator - based etof show that the multiple scattering effects , mainly from the mdc endcap materials , can cause multi - hit on the etof s readout cell and make the hit position distribution structure on the etof s readout cell complex by producing secondary particles in the materials and significantly influence the performance of etof .furthermore , the complex hit position distributions also indicate that the tracking accuracy is also worse in the endcap region than that in the barrel region , which consequently makes the position - dependent time calibration difficult .  to reduce the multi - hit probability and simplify the hit position distribution structure , smaller readout cell size is favored .however , reducing module size means increasing readout channels , which is not suitable for scintillator - based etof since the pm dedicated in strong magnetic field is rather expensive .the mrpc , first developed by lhc - alice tof collaboration , is a new type gaseous detector with good time resolution , high detection efficiency .furthermore , it can be designed with a smaller readout cell structure and the cost of each readout cell is much lower .it is considered a suitable candidate for the upgrade of besiii etof system .the r&d for such an upgrade began in 2010 .in each endcap there are 36 trapezium - shaped mrpc modules .fig.[7 ] shows the top and side view of an mrpc module appropriate for besiii .it has a double - stack structure with twelve gaps .floating glass sheets are used as the resistive plates .the thicknesses are 0.4 mm and 0.55 mm for the inner and outer glass , respectively .the gap between the glasses is 0.22 mm .the thickness of the honeycombs , readout pads , pcb and mylar boards are also shown in fig.[7 ] .the mrpc is placed in a aluminum box with a thickness of 1 mm which is filled with a standard gas mixture for mrpc .the component of the gas is 90% freon + 5% sf6 + 5% c4h10 .two types of mrpc modules with different readout pad structure are designed for the upgrade .the dual - readout mrpc module has 12 readout pads of 2.5 mm wide , with the length ranging from 8.6 cm to 14.1 cm .signals are readout at both ends of the pad .the single readout mrpc module has a similar structure except that each readout pad is divided into two from the center ( see fig.[3 ] ) . in order to study the improvement with this upgrade, the scintillator - based etof system is replaced with the proposed mrpc based etof system in the simulation setup shown at fig.[2 ] .the hit position distributions on mrpc based etof module , of primary electron , secondary electron / positron are shown in fig.[4](a ) and ( b ) respectively .the module boundaries are also shown in the figures like that of scintillator - based etof case .note in fig.[4](c ) the mrpc pad boundaries are not shown since mrpc are intrinsic not sensitive to gamma ( usual efficiency 0.1% ) .it s found that the multi - hit rate is 21.8% or 16.7% for mrpc etof with dual - readout or single - readout module design , dropping significantly from scintillator - based etof s 71.5% .note the readout cell of mrpc based etof is much smaller than that of scintillator - based etof and all three components in fig.[4 ] ( primary electron , secondary electron / position and gamma ) contribute for a scintillator while gamma hits are not accounted for mrpc .  with a significantly reduced multi - hit event rate and simpler hit position distribution structure, one can expect better timing performance and easier calibration for a mrpc - based etof rather than a scintillator - based etof . to compare the timing performance of both types of etof, we also make a semi - quantitative calculation for mrpc based etof in a similar way as the scintillator .  on the mrpc readout pad ,electric pulses propagate to the fee - fed end . for each track , the signal transmission time on the readout pad is calculated as l / v , where l is the distance between the hit and the feed - out end , and v is the propagation velocity of the electric pulse . for mrpc modules used in besiii etof , v-1 is measured to be 45 ps / cm , near twice faster than that in an etof scintillator .also in each event , only the earliest signal arrival time is taken as the measured tof . for single - readout mrpcthe signal is chosen to be feed - out from the left end , while for dual - readout mrpc , the time measured is chosen as the average measurement from both sides .the tof distribution of a mrpc etof is shown in fig.[8 ] .it s obvious that the false timing from secondary particles is greatly suppressed compared to the scintillator etof case .we make a comparison of timing performance from all three kinds of etof design ( two detector technology , and two readout pad design for mrpc ) and show the result in fig.[9 ] .again we see clear improvement with mrpc - based etof . for the two kinds of mrpc readout design , the single - readout method has better timing property due to less multi - hit rate compared to the dual - readout one .however , one should know for single - readout mrpc , the timing is hit position dependent thus requires precise tracking which is not easy at besiii endcap region .dual - readout mrpc basically does not need very good tracking since the hit position uncertainty cancels out in time averaging from both ends .further investigation need more detailed simulation and experimental efforts and is beyond the scope of this paper .      to further quantify the difference in timing for the three kinds of etof design, we calculated the fraction of events in which the time difference ( with respect to time=0 in figure [ 9 ] ) larger than 30 ps , 50 ps , 80 ps , 100 ps and 150 ps .the results are shown in table.[tabone ] .we can clear see that the upgrade can further reduce on the adverse influence of the multi - hit on the timing of etof .@*6l  types&&&&& & 40.0% & 36.2% & 33.3 % & 32.1 % & 30.0% + dual - readout mrpc & 16.3% & 11.6% & 5.9% & 3.3% & 0.5% + single - readout mrpc&9.4% & 7.1% & 3.7% & 1.9% & 0.0% +we have developed a geant4-based mc model to study the timing property of the two types of etof design - based on scintillator or mrpc .the simulation results show that multiple scattering process in the material budget between mdc endcap and etof will produce a large number of secondary particles that will affect the timing of the plastic scintillator used in current besiii etof system , causing a high multi - hit rate ( 71.5% ) and a double - peak structure in the tof spectrum for bhabha events .the timing peak with tof abnormally small is contributed by the secondary particles with very low energy and is about 0.9 ns earlier to the nominal timing peak .these results are consistent with the offline calibration results with experimental bhabha data .the multi - hit rate of mrpc is much lower ( 21.8% for dual - readout mrpc and 16.7% for single - readout mrpc design ) , and the timing performance of mrpc is also better .this work was supported by national science foundation of china , no.10970003 .

Abstract: a geant4-based monte - carlo model is developed to study the performance of endcap time - of - flight ( etof ) at besiii . it s found that the multiple scattering effects , mainly from the materials at the mdc endcap , can cause multi - hit on the etof s readout cell and significantly influence the timing property of etof . multi - gap resistive plate chamber ( mrpc ) with a smaller readout cell structure is more suitable for etof detector due to significantly reduced multi - hit rate , from 71.5% for currently - used scintillator - based etof to 21.8% or 16.7% for mrpc - based etof , depending on the readout pad size used . the timing performance of a mrpc etof is also improved . these simulation results suggest and guide an etof upgrade effort at besiii .   _ keywords _ : besiii , multi - hit , etof upgrade , mrpc , time resolution


mages are one of the major sources of information transmitted and stored in the digital media .they are widely used in satellite communication , social media , medical diagnosis , and other applications where visual content is required . in classical computers ,image processing is a well - developed field applying fourier transforms , wavelet transforms and other various transforms on images for different applications .likewise , the development of quantum fourier transform [ 1 - 2 ] and quantum wavelet transform [ 3 ] have enabled the development of image processing on quantum computers .however , to apply these transforms , images on quantum computer need to be suitably represented and stored .quantum computers , first proposed by richard feynman in 1982 [ 4 ] , has led to the advent of quantum computing . during the last two decades, extensive research has been conducted in quantum computing theory , resulting in the development of primier algorithms like grover s database search algorithm [ 5 ] and shor s integer factoring algorithm [ 6 ] .it has been proved theoretically that quantum computers perform operations much faster than classical computers . with the view of developing a quantum computer in the future ,the fields of quantum information processing , quantum information theory , quantum cryptography and quantum communication are also explored , and intensive research is being conducted in these areas [ 7 - 8 ] .quantum image processing ( qip ) is a new addition to the quantum computation and information theory .although qip is currently at a nascent stage , its full scale development would enable various multimedia applications pertaining to images and videos on quantum computers .venegas - andraca and ball [ 9 ] proposed a method for storing binary shapes in images on a quantum computer through quantum entanglement .although this paper provided a method for efficient and generic storage of geometrical shapes , it did not suggest any method to store the complete image .moreover , the paper only concentrated on binary images , whereas real life images possess multiple intensity levels .subsequently , le et al . [10 ] provided a flexible representation of quantum images for multiple intensity levels and different colors .it used the information of colors and its corresponding position to represent a pixel value .however , this representation can be restructured .a different set of parameters instead of the normalized color information can be used , which conforms better to quantum systems .in addition , the single dimension -qubits position identifier can be replaced with a two - dimensional multiple qubit position identifier .as the images are two - dimensional signals , a two - dimensional multiple qubit identifier would be more appropriate to represent image pixels .c|cccc & 1 & 2 & 3 & 4 +  + 1 & & & & + 2 & & & & + 3 & & & & + 4 & & & & +  c|cccc & 1 & 2 & 3 & 4 +  + 1 & & & & + 2 & & & & + 3 & & & & + 4 & & & & +  in this paper , we propose a new representation of images on quantum computers based on the location of pixel values and the normalized amplitude of the signal values of a gray scale or a color image .considering the fact that the gray - scale images or the different color channels of images are two - dimensional vector space of x ( row ) and y ( column ) , the method uses the dual representation of row - location vector and column - location vector , separately , to identify a pixel in an image .each location vector can be defined as the quantum state at row and column using multiple qubits .the direct product of the row - location vector and the column - location vector results in a two - dimensional quantum state of pixel in the liouvillian space .subsequently , the coefficients of different quantum states consisting of both row and column information are the normalized amplitude values of image signal .the normalization of amplitudes is derived from the constraint that the coefficients of the quantum states should be unit vector .the proposed representation is general and can be utilized to represent all types of color channels which represent color images for different applications . unlike other quantum image representation techniques and all known classical image representation techniques, the proposed representation provides a new approach of storing and transmitting information of locations in the image matrix where a particular amplitude value is occurs , instead of storing and transmitting the content or amplitudes in the image matrix .the following is the outline of this paper .the representation of pixel location / s using two - dimensional multiple qubits is shown in section ii .section iii of this paper deals with the normalization of image signal amplitudes . in section iv , a method for generic representation of gray - scale and color imagesis given by merging the normalized amplitudes of the image signal with corresponding two - dimensional multiple qubits pixel location . in the end ,section v discusses the advantages and limitations along with the future scope of this work in the proposed quantum image representation .one of the important features of this paper is the representation of a two - dimensional image by row and column vectors .each pixel location in a classical image is specified by a row and a column number .similarly , any pixel can also be defined by row and column vectors on a quantum computer .this can be achieved by generating -length row - location vector and -length column - location vector with -qubits and -qubits , respectively . for row vector, -qubits can be generated with orthonormal binary qubits which are defined as ,   the above binary qubits represent the constituent vectors for row location . the notation is called ket .the recursive tensor products of the various permutations of these constituent vectors form -length row - location vector using -qubits .mathematically , it is defined as , where , is the row - location vector or the state of -qubit , , and is the row number of pixel .  likewise ,orthonomal constituent vectors for -length column - location vector can be written as ,      here , notation is called bra . similar to the row - location vector , the column - location vector is also generated by taking the recursive tensor products of the constituent vectors using bra notation , where, is the column - location vector or the state of -qubit , , and is the column number of pixel .  to represent a pixel location in its complete two - dimensional format , the outer product of the row - location vector and the column - location vector is taken .hence , the two - dimensional location of a pixel can be identified by the following , where , is the pixel at row and column .  to illustrate with an example ,the pixel location of any image matrix in terms of two - dimensional quantum states is given in . as can be seen, expands equation 7 for each pixel location representing bra part as a -qubit quantum state of the row - location vector , and ket part as a -qubit quantum state of the column - location vector .furthermore , through , the orthonormality of each quantum state can also be derived .it should be noted that is another way of representing . uses the binary represention of the quantum states , whereas uses the decimal representation .the reason for showing is that it occupies less space and would be used in section iv for a compact quantum image representation , instead of .digital images are quantized analog images .the number of quantization levels depends on the total number of bits used to represent all possible values . in a gray scale image ,quantized values represent intensity or luminance of the image . on the other hand ,color images are expressed by different color spaces , like red - green - blue ( rgb ) , hue - saturation - value ( hsv ) and luminance - chrominance ( yuv ) , to name a few , depending on the application . in the case of a gray scale image ,the amplitude of the image signal is the intensity value , but the same might not be true for the different color channels of various color spaces .for example , chrominance color spaces in yuv color space contains normalized color difference between the two color channels of rgb color space .hence , considering various color spaces , the normalized intensity value proposed in [ 10 ] can be generalized with the normalized amplitude value of digitized image signal .  in quantum computation theory ,the scalar amplitudes ( ) of the quantum states are complex numbers ( ) . like the quantum states and the superposition of quantum states, these amplitudes are also constrained to be a unit vector , i.e.    or in the matrix representation ,    however , in the case of any color channel of the image , amplitudes are always on the real line .furthermore , in most color channels , they lie on the positive real axis . thus , the following can be stated without the loss of generality ,    now , let us consider the digitized image signal - whether gray scale or any channel of color image - a two - dimensional matrix of size , having some value , called amplitude , at each pixel location .let be the amplitude of the digital image signal at row and column , where  ] .after incorporating the constraints of equations 8 and 9 , the normalized amplitude image matrix ( ) can be defined as ,    or ,    continuing with the example in section ii , shows the expansion of the normalized amplitudes ( ) of image matrix ( ) in equation 12 for the image size .c|cccc & 1 & 2 & 3 & 4 +  + 1 & & & & + 2 & & & & + 3 & & & & + 4 & & & & +  after achieving the normalized amplitudes of the image signal in equation 12 , the next task is to represent these amplitudes also by multiple qubits quantum states .the reason for this approach is to reduce the normalized amplitude values to quantum states so that the amplitudes can easily be stored and transmitted in a quantum environment .also , this reduction to quantum states allows storage and transmission directly at machine level .  for typographical convenience ,the numerator and the denominator in equation 12 are replaced as and , respectively .now , equation 12 can be rewritten as ,    both , and are real values , and can be integer or fractional values . in either case , the number of qubits required to represent these values will be and for the numerator and the denominator , respectively .thus , the numerator and the denominator of the normalized amplitudes is represented by qubits in the following way :      c|cccc & 1 & 2 & 3 & 4 +  + 1 & & & & + 2 & & & & + 3 & & & & + 4 & & & & +  c|cccc & 1 & 2 & 3 & 4 +  + 1 & & & & + 2 & & & & + 3 & & & & + 4 & & & & +  as can be seen , the denominator in equation 13 is constant for all the normalized amplitudes of a particular color channel or image .hence , it can be segregated and represented separately .the qubit representation of the numerator expressed in equation 14 is shown in for the image size .unlike the qubit representation shown in , the quantum states at different pixel locations can be same .this is due to the fact that the amplitudes of the image signal are completely independent of each other , and the pixel locations . on the other hand ,the quantum states of the pixel locations are mutually exclusive of each other .another fact which can be observed from equation 14 and is that the quantum state representation of both the numerator and the denominator are individually unit vectors , conforming to the constraints of quantum computing theory .after calculating the normalized amplitude of the image signal , and deriving each location of two - dimensional images , the quantum images can be represented as the superposition of the products of the normalized amplitude ( numerator ) and its corresponding two - dimensional quantum state .the denominator is stored separately to avoid redundancy of information and extra storage space .the following equation defines the proposed quantum image representation mathematically ,    where is a two - dimensional gray scale quantum image , or one of the color channels of the quantum color image . after substituting from equation 7 ,the above equation is rewritten as ,   the above equation is further simplified by using equations 3 and 6 to replace , and equation 14 to replace , to be written as ,    or by substituting and with and , we can write    in , each expression of equation 19 for size image matrix is shown with respect to its pixel location .it can be seen that each pixel contains the normalized amplitude ( numerator ) of image signal and a two - dimensional quantum state corresponding to pixel s location .the proposed representation focusses on two features of images , the amplitude of the image signal and its corresponding pixel location on the image space . the main idea behindthis representation is to decorrelate both the features mentioned earlier .this is due to the fact that for a given or fixed image dimension , pixel positions would be identified using the same quantum states of the row - location vector and the column - location vector .thus , only information which needs to be stored or transmitted in quantum systems would be all the pixel locations corresponding to the same amplitude value .moreover , this representation is also able to decorrelate individual amplitude values with its neighborhood amplitudes .it may initially seem to be a disadvantage of this representation because the image signal amplitudes are highly correlated and this correlation produces high image compression .but it should be observed that instead of storing and transmitting amplitudes of the image signal , what actually stored and transmitted are all the pixel locations having the same amplitude values , for all the amplitudes present in the image .one reason for this approach is that the two different amplitudes can not occur at the same two - dimensional pixel location .hence , once a position has been allocated to an amplitude value , it would not be used again for other amplitude values resulting in the reduction of information to be stored or transmitted by introducing the conditional entropy .the second reason for this representation is that for a given image dimension , all the pixels would have fixed qubit location irrespective of the image content .therefore , the things that would change during the storage and transmission of different images are the number of locations occupying a particular amplitude and its respective two - dimensional quantum state of column and row .  there are two major advantages of this representation : first , it is independent of bit depth of input image , and second , it is valid for any color channel of any color space which represents an image , in part or whole . as mentioned earlier in the paper , a quantum computer would store all the two - dimensional quantum states for a particular amplitude , and hence , increasing the bit depth would only increase the number of amplitudes for which locations have to be specified . the total number of locations would always remain constant for any fixed image dimension . in the case of other representations ,more storage space is required to represent an image , because the number of bits needed to represent each pixel is directly proportional to the bit depth .in addition , the proposed representation only requires information for the amplitudes present in an image and not all possible amplitudes generated by bit depth .the second major advantage of comprehensive representation for any color channel is due to the consideration of the amplitude of color channel representing the image signal as the coefficient of the joint quantum state of row and column vectors . until now ,all the quantum image representation techniques considered rgb color space for representation while excluding other colors spaces .there are many color channels , like chrominance in yuv , which do not store the luminance of an image , or the luminance of different colors which are combined to form an image .however , these color channels are quantized to a fixed number of values so that they can not possess any value ranging in the set of real numbers .these fixed number of values act as amplitudes for the respective color channel in the proposed representation resulting in the generalization for all the color spaces .in addition to the two major advantages , the proposed representation provides primitive level security to the image . to elaborate, the original image can not be reconstructed successfully until the denominator ( ) of the normalized amplitude is available .this is because the normalized numerator value can have many valid reconstructions .the denominator ( ) acts as a key which results in the desired image .hence , applications like watermarking , primary level cryptography , and password protection , to name a few , can be inherently used in this representation .nonetheless , there are a couple of limitations in the presented quantum image representation .first , the requirement for the number of qubits increases with the increase in image dimension resulting in more storage space per pixel compared to the bit depth per pixel in other representations .second , the number of qubits required for identifying the quantum state of the row - location vector or the column - location vector depends on the ceiling of the logarithmic value of the length of each vector .this leads to extra fractional bits per pixel ranging between 0 and 1 .these limitations lead to future work .it is suggested that the limitations can be overcome by employing effective lossless compression technique on the qubits representing quantum states .the authors would like to thank dr .r. atre ( department of physics , jaypee university of engineering and technology , india ) for providing valuable inputs on quantum computing .a. klappenecker and m. rtteler , `` discrete cosine transforms on quantum computers , '' in _ proc .ieeer8-eurasip symp . on image andsignal processing and analysis _ ,croatia , 2001 , pp .464 - 468 .c.c . tseng and t.m .hwang , `` quantum circuit design of 8 8 discrete cosine transforms using its fast computation flow graph , '' in _ proc .ieee int . symp . on circuits and syst ._ , japan , 2005 , vol .828 - 831 .le , f. dong and k. hirota , `` a flexible representation of quantum images for polynomial preparation , image compression , and processing operations , '' _ quantum inf . process .63 - 84 , 2010 .

Abstract: in this paper , we have proposed a novel method of image representation on quantum computers . the proposed method uses the two - dimensional quantum states to locate each pixel in an image matrix with the normalized amplitude values of the image signal being coefficients of the quantum states . the two - dimensional quantum states are linear superposition of outer product of -qubits row - location vector and -qubits column - location vector of each pixel location in the image matrix . in classical systems , these quantum states are two - dimensional binary matrices having 1 at the targeted pixel location and 0 at the other pixel locations . the fact that linear superposition , tensor product and qubits form the fundamental basis of quantum computing , the proposed method presents the machine level representation of images on quantum computers .   shell : bare demo of ieeetran.cls for journals  quantum image processing , image representation , quantum computing , linear superposition , tensor product .