the concept of folding has an important place in polymer physics  . considering a polymer chain made of identical constituents ( the _ monomers _ ) ,the entropy of such a system can be obtained by counting the number of inequivalent ways of folding the chain onto itself .if the model of polymer does not take self - avoidance into account , it is then equivalent to the well - known brownian motion .several more involved models have been proposed which study the _ compact _ folding of a _ self - avoiding _ polymer as a hamiltonian cycle ( i.e. a closed , self - avoiding walk which visits each vertex ) on a regular lattice .they can also be defined for some kinds of _ random _ lattices  , where each configuration is now described by a system of non - intersecting arches which connect the pairs of monomers , which are neighbors in the real space .  in the present paper , the compact folding of a polymer chainis modeled by a folded strip of stamps , with a complete piling of the strip on top of one stamp  .it is then equivalent to another model of non - intersecting arches , the so - called _ meander _ problem , which can be summarized by this simple question : in how many ways can a road cross a river through bridges , and possibly wind around the source .a related problem can be defined by forbidding the winding around the source ( i.e. the river is infinite at the both ends ) .it is now equivalent to enumerate the `` simple alternating transit mazes ''  of depth ; it was also investigated in connection with hilbert 16th problem , namely the enumeration of ovals of planar algebraic curves  .  by analogy with some models of statistical mechanics likerandom walks or self - avoiding walks , the meanders can be described in the language of critical phenomena . in particular , the asymptotic behavior of meanders when is large can be characterized by `` critical '' exponents . however the exact enumeration of meanders is particularly complicated : there is no known formula for in terms of . by generating all possible configurations , by hand or with a computer , the beginning of the sequence be computed  exactly .as increases exponentially with , the limits of computers are reached for and the estimates of the exponents are too inaccurate to validate ( or invalidate ) some conjectures .one should mention that several exact results , for arbitrarily large , which are unfortunately not helpful to determine the values of the exponents , have been obtained with other techniques : random matrix model methods  and an algebraic approach using the temperley - lieb algebra  or the hecke algebra  .many models in statistical physics can be studied by monte - carlo ( or stochastic ) methods . with these algorithms , only a small set of configurations among all the possible onesare generated . in principle , the expectation of physical quantities ( like energy or magnetization ) with a given law of probability ( like the boltzmann law involving an external temperature ) can be estimated from these randomly generated samples , if their probabilities of generation are known .for example , with the metropolis algorithm for classical spin systems  , the probability of generation is built to be equal to the boltzmann law and the average is done over the generated configurations with equal weights . to bypass some difficulties ( for example when the phase space has many local minima with high free energy barriers between them ) , it is possible , in principal , to generate the random configurations with another more adapted law and to correct this bias when the average is done  .but for the meanders problem , the situation is quite different : the phase space is not easy to built because the number of configurations is unknown and the only known efficient method to draw a meander of size is a recurrence over .moreover the nave way to use this recurrence gives a distribution of meanders which is not flat , and this default increases exponentially with .this paper presents a monte - carlo method which explores the meanders with an almost flat distribution law .furthermore the bias is known and can be corrected exactly .therefore , the average can be done over meanders with equal probabilities .in particular , better estimates of critical exponents are obtained .after section 2 devoted to the definitions , and section 3 which explains the building of meanders by recurrence , section 4 of this paper describes the monte - carlo method .the results are presented and discussed in section 5 .a _ meander _ of size is a planar configuration of a non - self - intersecting loop ( _ road _ ) crossing a half line ( semi - infinite _ river _ with a _ source _ ) through points ( _ bridges _ ) .two meanders are considered as _ equivalent _if their roads can be continuously deformed into each other , keeping the bridges fixed : this is therefore a topological equivalence .we call an _ arch _ each section of road between two consecutive bridges .so a meander of size has bridges and arches .the number of different meanders of size is denoted by .for example , , , , . in fig .[ m4 ] , the 4 meanders of size 4 are drawn .the s , up to , can be found in ref . .in previous articles  , these objects were called _ semi_-meanders , to distinguish them from the case where the line is infinite ( river without source ) . in this paper ,the river is always a half - line and the word _ meanders _is used for convenience .we can define  meanders with _ connected components _, i.e. made of one river and non - intersecting roads .but , in this work , we do not include this generalization and we keep .however the monte - carlo method , used in this article , can be adapted without difficulties to an arbitrary fixed , and even for varying with a fugacity .  as explained with many details in ref . , the meander problem is absolutely equivalent to the problem of the compact folding of a strip of stamps because each meander of size can be continuously deformed in such a way that the `` road '' becomes a vertical line and the `` river '' becomes a folded strip of stamps .we prefer to present our results with the meander representation because the main recursion relation , described later , seems more `` natural '' in this picture .the meander problem has certain similarities with two - dimensional self - avoiding walks : a meander is obtained by intersecting a closed self - avoiding walk by a half - line and keeping only the topological aspect . by analogy , it is expected  that where the estimates given in ref .are and .the connectivity can be reinterpreted as the average number of ways of adding a bridge close to the source by deforming an arch of a given large meander .then is the entropy per bridge .the configuration exponent is sensitive to the boundary conditions , for example whether the road is closed or open , whether the river is infinite or semi - infinite , straight or forked , whether the meander is drawn on a planar surface , a sphere or a surface with higher genus .conversely , we expect that remains the same for all these boundary conditions .it is similar in on - lattice self - avoiding walks problem where the connectivity depends on the type of lattice ( square , honeycomb ) and not on the boundary conditions , whereas the `` universal '' configuration exponent depends on the boundary conditions , but is not sensitive to the small scale details of the lattice . for these reasons, we think that the numerical value of is valid only for this particular model of meanders . but is expected to be more `` universal '' and to keep its value in other variants of the meander problem .unfortunately , the numerical determination of is less precise than , because describes the correction to the leading exponential asymptotic behavior .  for a given meander , the _ winding _ number of the road around the source of the rivercan be defined as the minimal number of intersections between the road and a half ( semi - infinite ) line starting at the source and extending the river on the opposite side . for an example , see fig .we define as the average of the winding number over all the meanders of size .we can see the winding number as the topological end - to - end distance between the source ( right end of the river ) and the infinite ( left end of the river ) . herethe distance between two points is simply the minimal number of arches which must be crossed to go from one point to the other . by analogy with the end - to - end exponent of self - avoiding walk, we expect that where the estimate given in ref .is .if we study the meanders by leaving free the number of connected components , the problem is equivalent  to a random walk on a semi - infinite line and can be studied with usual methods of combinatorics . in particular , , and is the brownian exponent .but , by fixing , the problem is drastically more difficult and the above values are , to our knowledge , not yet known exactly .  for a given meander of size ,we label by each segment of river in - between two consecutive bridges , from right to left .so the rightmost segment ( with source ) is labeled 0 , and the leftmost ( semi - infinite ) segment is labeled .we define the _ height _ as the number of arches _ over _ the segment , and as the number of arches _ below _ the segment . an example is given in fig .[ haut ] .for the case , the both definitions and are equivalent and equal to the winding number : . from the definition, we have , and .  for a given meander of size , we define the _ area _ as for meanders of size , it can be proved that the maximal area is for the two meanders with a snail shape ( where plus the symmetric meander ) , and the minimal area is ( resp . ) when is even ( resp .odd ) for the ( resp . ) snake shaped meanders characterized by for . as in the case of the winding number, we will consider the _ average _ profile height and the _ average _ area over all the meanders of size .in this section , we describe a recursive algorithm to enumerate and built all meanders of a given size . though it was described in ref ., we prefer to recall it in details , because the monte - carlo method is based on this recursion .we have different ways to built a meander of size , starting from a meander of size .our method consists in adding a bridge on the left most part of the river ( opposite to the source ) and changing the road to cross this new bridge . to keep this change minimal , only an _ exterior_ arch is modified ( an arch is _ exterior _ when no other arch surrounds it ) .take a meander of size ( the _ parent _ ) and choose ( or label ) one of its exterior arch . by the process described in fig .[ recur ] , a meander of size ( the _ child _ ) is built .a parent has as many different children as exterior arches . by inverting this process, it appears that each meander of size has one and only one parent .more precisely , it is a one - to - one mapping between the meanders of size , and the meanders of size with one labeled exterior arch .the starting point of the recursion is the unique meander of size 1 .by successive applications of the recursion process , every meanders of size can be built .as shown in fig .[ arbre ] , the set of meanders is organized as a _ tree_. the root , at level 1 , is the starting meander .each branch between a node on level and a node on level represents a relation between a parent of size and its child of size .apart from , a meander ( or node ) has several exterior arches , then several children ( or branches ) .their number depends on the precise shape of the parent and varies between 2 and .= 8.5 cm  if we want to exactly enumerate the meanders of size , the only method we know is to built and investigate the tree up to the level . in particular, we have not found a direct recursion between the numbers .the number of children of each meander has a distribution which seems to be erratic and the only way to know it is the examination of its shape .then , to compute , the work is proportional to . as the s increaseexponentially , the limits of the capabilities of the computers are rapidly reached .  in ref . , the meanders numbers up to are given .with a recent computer and more tricks of programming , it is perhaps possible to obtain or 32 .if the power of computers continues to increase exponentially , the best we can expect with the full enumeration without major improvement , is a linear growth in , with a rate of only one new size every 2 or 3 years .we have the feeling that this progress is too small to change significantly our understanding of the meanders problem .  in this article, we will investigate larger with a monte - carlo method increasing more slowly than an exponential .but , this stochastic method gives results with error bars and the interpretation is more delicate .as explained in the previous section , the exponential growth of the computations with an exact enumeration method limits the size of meanders around . to reach bigger , it is then natural to try to study this problem with a monte - carlo ( or stochastic ) method .as the set of the meanders of a given size is too large to be fully explored , the general idea is to randomly select a small subset .then , the measurements are done and averaged on the selected meanders .it gives an _ estimator _ of the exact ( but unknown ) result , with a unknown error .this error has two components : statistical fluctuations and bias .the statistical fluctuations can be reduced by independently repeating the procedure many times .then , we obtain a histogram of the estimator , with an average and a variance . under the hypothesis of finite variance, the statistical fluctuations of the average can be estimated by usual formulas of statistics .the _ bias _ is the difference between the exact result and the mathematical expectation of the estimator .if it can be exactly calculated , we subtract it from the estimator .but , in general , an unknown part remains , which can not be reduced by a better statistics .as explained below , by adjusting parameters of the simulation , the bias can be reduced to become smaller than the statistical fluctuations .  in this section, we will first introduce the simplest algorithm , the _ one - squirrel method_. we will see that its statistical fluctuations grow exponentially with and they are too big for . then , we present an algorithm , the _ multi - squirrel method _ , for which the fluctuations increase less rapidly .the method is based on the recursion relation ( see fig . [ recur ] ) , with which the set of meanders is organized as a tree ( see fig . [ arbre ] ) .the monte - carlo squirrel has the following stochastic behavior .it starts at the root of the tree ( the meander of size ) .it climbs into the tree . at each level , it stands on a node and makes some measurements concerning the meander of size , represented by this node .then the squirrel goes to the level by choosing at random one of the branches , starting from this node .the squirrel stops at a prefixed level .this process constitutes _one _ simulation .the probability that the squirrel reaches a given meander of size is .this probability law is not flat because the sequence of depends on the visited nodes . as seen in fig .[ arbre ] , for , the two meanders on the left side have a probability and the three on the right side have .so , to correct this bias between meanders , the squirrel has a weight which is calculated during its climbing .  by noting the mathematical expectation ( over all possible simulations ) , it is then obvious that , because the sum runs over possible paths and the contribution of a given path is with probability .  more generally , for some quantity ( for example the winding number ) , we wish to determine where is the value of on the -th meander of size .each simulation gives , at level , a measurement on the meander reached by the squirrel and which is the generalization of obtained with .then is an _ unbiased _ estimator of ( i.e. ) .  as usual in monte - carlo methods ,several simulations are made independently and we hope that the average of all the measurements is close to . unfortunately , this method does not work in practice , because the weight is the product of .although the distribution of each is regular , the product of many random variables is not self - averaging .as the sum of is self - averaging ( i.e. the observed result is closed to its mathematical expectation when goes to infinite ) , most of the observed are not closed to and increases like an exponential .then the averages with weight are dominated by exponentially rare events and the statistical fluctuations become large . to keep the observed average close to the mathematical expectation , the number of simulations must increase exponentially with and fluctuations become too big for or 35 .as the difficulties increase exponentially with ( as for exact enumeration ) , is is useless to increase the power of the computer .we need a new algorithm .we generalize the one - squirrel method , but now with a population of squirrels , which reproduce and die ; is a fixed parameter during all the simulations .it is more simple to choose as , with at the starting point , a squirrel staying at each node of level ( meanders of size ) . in this work , and we use the up - down symmetry to reduce the population to squirrels .the population evolves from level to level by the following process .each squirrel lives on a node on level , connected to nodes on level .it reproduces and has children and each child lives on each one of these nodes .the total number of children is calculated .the ratio is an estimate of : the average of the number of children per parent .to prevent an exponential growth of the population and of the needed computer memory and time , the total population is keep constant by decimating the children : only among the children survive .the choice is made at random with uniform distribution .then the probability of surviving is .this decimation is the single monte - carlo step of the algorithm .this process is iterated up to reach a prefixed level : it gives _one _ simulation .many independent simulations are done and averaged .  the particular case gives the previous method with one squirrel .but , as for , for every value of , the probability that a given meander is reached , is not uniform .the nodes with small number of `` brothers '' or `` cousins '' have always a small advantage . butthis bias becomes smaller when the population is large .that is the main improvement of this method .the limit corresponds to the exact enumeration .  to correct the bias ,each simulation has a weight and the averages runs over all the simulations with their weight .more exactly , for some quantity , by keeping the notations of eq .( [ aa ] ) , one simulation with squirrels gives measurements for $ ] with a weight and the estimator is unbiased , i.e. we note that the case gives .in order to prove eq .( [ moy ] ) , we define the operator characterizing a given meander by when and 0 otherwise .then every operator can be split up into . as the expectation of a sumis always the sum of expectations , we have to prove eq .( [ moy ] ) for the operators only , which becomes , where if the -th meander is occupied by a squirrel and 0 otherwise .let represent the parent of in the tree at level .the probability that ( i.e. is occupied ) is the product of the probability that was occupied at the level and the probability that its child survives after the decimation process .  by using eq.([qn ] ) and averaging on the random decimation only , it is a recursion relation between a given meander of size and its parent of size . by iterating , we go down to the ancestor at the starting level for which and the empty product of is 1 .it proves that for every meander of size , and eq .( [ moy ] ) is valid for every operator .  as usual, many simulations are done and the measurements are averaged with their respective weight . a priori , it seems that this method has the same defect as the _ one_-squirrel method because the weight is the product of many , not self - averaging when becomes large .the ratio between the mathematical expectation and the most frequently observed increases like an exponential .but , the main improvement of the _ multi_-squirrel method is that the distribution of becomes narrow when the population of squirrels is large . as ,the fluctuations of are of order ) because the number of children is the sum of random variables .a taylor expansion of shows that   then , with these simple arguments , we can hope that the ratio ( [ ratio ] ) grows like and that problems appear only when becomes on the same order than . in our simulations , we observe that the fluctuations grow with faster than this optimistic prediction . in fact , the s are not independent and the exponential function accentuates all deviations .then we supervised carefully the distribution of .when is small , we see a regular bell - shaped curve .but , when increases , the distribution becomes asymmetric , with a long and irregular tail for the large .for example , for , with squirrels , the width of the distribution is only 12 % , but we observed rare events with a value of as big as three times the average .however , in this case , their contribution to the average and fluctuations is not yet problematic .but , if we let increase without control , rare events will dominate and the results will become hazardous .  how to choose and ?the nave point of view is to take as bigger as possible .but , to make independent simulations with squirrels of size , the need for computer memory is of order and the need for computer time is of order . if is large enough , the fluctuations are gaussian and the error bars are of order .as is always limited , we will extrapolate to study the asymptotic behavior .for that , it is of no help to have large values of if the error bars are too big .so for a fixed computer time , we prefer accumulate good statistics by limiting . finally for a fixed product , we prefer to take as bigger as permitted by the memory computer to avoid the problem of rare but large fluctuations .   in the previous section ,we have seen how to obtain unbiased monte - carlo estimates of the sum over all the meanders of size of some quantity ( see eq .( [ aa ] ) and its notations ) .however we are more interested by the _ average _ over all the meanders of size . for example , the average winding number ( see eq .[ wn ] ) is obtained when counts the winding . to evaluate of eq .( [ rnng ] ) , we can analyze ; in this case , counts the exterior arches .more generally , we want to use non - linear combinations of and .  with our monte - carlo method, we have seen that one simulation gives a measurement which is an _ unbiased _ estimator : . with independent simulations , we call the usual average of the measurements ; its fluctuations are times smaller .the bar over the symbols distinguishes the average of observed values by monte - carlo method , from the ( unknown ) mathematical expectation , marked with .the same work can be done with which is a unbiased estimator of .we must be careful with the monte - carlo estimate of .for example , the average of the ratio gives bad results .it is better to compute the ratio of the averages .indeed , with a taylor expansion of and around their mathematical expectations and , the bias ( defined as the difference of the mathematical expectation and the target ) it can be neglected if it is smaller than the stochastic fluctuations .usually , in monte - carlo simulations , this problem disappears because several millions of independent measurements are done .but , in this work , the situation is quite different .in fact , each simulation is a complex process involving millions of squirrels , and the number of simulations is small .  of course, we can not compute this bias exactly , otherwise we would have already subtract it from measurements .but we can estimate it by the following process .the set of simulations is divided into subsets , with simulations each . in each subset , is computed .we obtained independent values , one for each subset ; by usual formulae of statistics , we compute their average and the error bars .this work is done for all integer between ( each subset contains only one simulation ) and ( only one set with all the simulations ) .which value of is the best ? following eq .( [ aqn ] ) , the bias of is expected to decrease like . for small values of , we observe really a dependency of on : the bias is visible .but , for , variations become smaller than statistical error bars : the size of subsets is sufficiently large to neglect the bias .but , if is close to its maximum , the number of subsets becomes very small and the error bars are not properly estimated . as in our work , we finally keep : the estimator is computed with 64 independent subsets of 128 simulations each .this method was used for all quantities presented below .it is valid , not only for ratios like , but also for non - linear functions like .it could also be possible to use more complex estimators .for example , the combination cancels the order of the bias .in this section , we describe the results obtained by our monte - carlo multi - squirrels method .after several tests , we used a population of squirrels for meanders with size up to . to do independent simulations , we have used during 8 days a parallel computer ( the cray t3e of the cea - grenoble ) with 128 processors ( dec - alpha at 375 mhz ) and 13 gigabytes of total memory , equivalent to 24000 hours of workstation cpu time .we have verified that the results are stable when ( the population ) increases .more exactly , the tests with smaller have larger error bars , but are compatible with results and error bars obtained with the largest . as explained in the sect .[ msm ] , we have carefully checked that is sufficiently large to explore sizes of meander up to .we want to measure and , which describe the asymptotic behavior of the number of meanders for large .the entropy can be estimated by .but it appears that the sub - sequences and have an alternating sub - leading correction .we have estimated it to be with .this alternating effect is dramatically amplified by the ratio .so it is better to consider with a jump from to .but even with this precaution , the reader can still see on the following figures a small parity effect . to estimate ,we have used the procedure described in sect .[ secbias ] .as we expect for large , by plotting versus , we can estimate ( limit when goes to 0 ) and the exponent ( asymptotic slope ) . in fig .[ lnr ] , we have plotted the monte - carlo estimate of versus for between 50 and 400 .we have arbitrarily subtracted the _ linear _ function , to reduce the amplitude of ; we obtain a figure where the small quantities ( remaining slope ) , ( limit when goes to 0 ) and curvature ( deviation to the expected linear behavior ) are more visible .the curvature remains small and a linear extrapolation gives a limit between and with an estimated slope . then   with the assumption that the asymptotic behavior is , the conjecture is incompatible with these simulations .but , we can try another asymptotic shape , for example by introducing a new exponent . in fig .[ lnrlog ] , we have plotted ( as in fig . [ lnr ] ) , but now versus . with this transformation of the -axis ,a linear behavior corresponds to and the slope measures .a linear extrapolation gives with compatible with the simple fraction .how shall we choose between both results eq .( [ b1 ] ) or eq .( [ b2 ] ) ?we notice that the quality of the alignment of points is the same in fig .[ lnr ] and fig .[ lnrlog ] .in fact , it is very difficult to distinguish between a logarithmic law and a power law with such a small exponent , with statistical error bars and when the amplitude of is small .in fact , for close to , the term looks like with .then , for any choice of not too large , gives a class of acceptable behaviors .we have tried to use more sophisticated extrapolation methods .for example , with a fixed jump , gives theoretically the same limit but by removing the term .thus gives a direct estimate of .unfortunately these kinds of derivative amplify statistical errors , and the results are compatible but less precise than the previous estimates .so , with our numerical simulations , we can not say if a logarithmic factor is present or not .however , , and we can exclude and the conjecture without logarithmic factor ( ) .   we will present the monte - carlo results of the exponent , which describe the asymptotic behavior of the average winding number . to avoid problem with bias , we have used the procedure described in sect .[ secbias ] . by plotting versus , the asymptotic slope will be a measurement of .[ fnu ] , we have plotted versus .we have arbitrarily considered and not because is less sensitive than to the finite size effects  . as the main question is to know whether or not , we have arbitrarily subtracted the linear function .then the variation of is reduced and we obtain a figure where ( residual slope ) and the curvature are more visible .we see that the curvature is small and a linear extrapolation gives . as it is difficult to estimate the errors with the data of fig .[ fnu ] , we have also tried more sophisticated quantities like which are discrete derivatives of with step .they give a direct value for , but unfortunately the statistical fluctuations are amplified by this differentiation and the uncertainty over is of order .= 8.5 cm  we have seen that , for the exponent , a behavior with logarithmic correction is not excluded by the monte - carlo data .so we have tried to fit the winding number with with this hypothesis , a plot of , as in fig .[ fnu ] , but now versus would give a straight line with slope . butthe curvature is much stronger than that of fig .so we dismiss this hypothesis and conclude that   with the assumption that the asymptotic behavior is a simple power - law , the brownian value is incompatible with these simulations .we define the probability distribution of winding number as the fraction of the meanders of size with windings .we expect  the asymptotic scaling behavior with a scaling function , where is the average winding number . with the factor 2 ,the integral of is normalized to 1 because and are integers with the same parity . in fig .[ pnw ] , we plot versus for different values of . to define the scaling variable , we prefer to take instead of to reduce finite size effects .= 8.5 cm  we see that the points accumulate on a smooth curve , which represents the scaling function . by analogy with the end - to - end distribution for polymers ,we expect  a power law behavior , , for small , and a behavior , for large .our data give the estimates to obtain a better precision , it would require to have larger values of .we are interested by the asymptotic behavior , for large size , of the average area ( see eq .[ av.area ] ) and average height ( see eq .[ av.height ] ) .= 8.5 cm  the label is the `` horizontal '' coordinate and varies between and .so , we introduce the scaled variable , with .we expect  that for large by fixing , where is a scaling function with integral normalized to 1 . in fig . [ profile ] , we have plotted versus for various size . only the positive are shown because is symmetric after summing over all the meanders .more precisely , we have plotted the average of over the monte - carlo samples of meanders , which is equivalent to the average of over these samples , plus those obtained by the left - right symmetry ( ) . as in the previous figures , we prefer to take and to reduce finite size effects .we see that , for , the points accumulate on a smooth curve , which represents the scaling function .this shape is not a half - circle , as it would be for a random - walk on a semi - infinite line  .as is the winding number for the particular case , eq .( [ rho ] ) can be valid only if scales as with the same exponent for all .consequently , the area scales as . in a previous section ,we have numerically determined by extrapolation of .the same work with and gives for both , compatible with the previous estimates , but less precise because the finite size effects are stronger .  in fig .[ profile ] , small deviations appear between the curves for =40 and 400 , at the boundary ( ) and in the middle of the meander ( ) .= 8.5 cm  in order to understand why finite - size effects are important near the boundary , fig.[bord ] is a plot of versus , with various .clearly , when is large , curves accumulate on a limiting curve   as this function can be fitted by a straight line on this `` log - log '' plot , we define a new exponent by when is large . as by construction , and for all , and exactly eq .( [ htilde ] ) .  if we set , by using eq .( [ htilde ] ) valid when is fixed , we obtain . on the other hand , by using eq .( [ rho ] ) which is valid when is fixed , we obtain now . as the exponents and differs , these two regimes are incompatible and the behavior of depends on the order in which and go to infinity .in particular , the domain of validity of eq .( [ htilde ] ) is reduced to the single point for eq .( [ rho ] ) .that explains why the rightmost dots ( ) in fig . [ profile ] for the small size are not superimposed on the curve obtained for the large size .  here , the exponent is the _ surface _critical exponent , while is the _ bulk _ critical exponent . near the boundary, is small and the condition limits appreciably the fluctuations toward the bottom .this effect is so strong that the exponent is changed and .this situation is reminiscent of other critical phenomena  , like the self - avoiding walk near a surface .our result is to be contrasted with the case of a random walk on a semi - infinite line for which the surface exponent keeps its brownian value .= 8.5 cm  finite - size effects observable near the middle of meander ( ) can be explained with fig .[ cusp ] , which is a plot of versus for small with various .clearly , when is large , curves accumulate on a limiting curve   if we make the hypothesis that the behavior of is compatible with eq .( [ rho ] ) by inverting the limits large and small , the consequences would be that has a cusp at with a infinite derivative when is small , and .but this power law behavior of is not observed .then the asymptotic behavior of with fixed and large is given by eq .( [ rho ] ) , i.e. , plus finite corrections of order .this cusp is an another boundary effect since the point is the source of the river .let us consider three consecutive heights for a given meander . by definition , .then , for a generic , the couple of its neighbors can have 4 respective values .but , for the special case , the situation happens only if a single arch connects the first bridge ( near the source ) to itself , by drawing a little circle around the source , without visiting the other bridges .this is forbidden if we insist in having only one connected component .in other word , the neighborhood of the source limits the fluctuations of toward the top .in particular , for every meander , .that explains why , on average , .  to understand why , it is mandatory to consider more complex forbidden situations for , , , , .for example , a circle connecting bridges 1 and 2 with an upper and a lower arch , with , is forbidden .more generally , for a given , the presence of the source forbids the systems of arches connecting the bridges with a closed road without visiting the other bridges .qualitatively , it is a repulsive `` force '' which favors the connection of bridge with bridges and gives a concave shape for small .  as the forbidden situations are more and more complex when becomes large , their statistical effects decrease and this repulsive force has a finite range . in the end, the summation over all forbidden situations gives finally this cusp with a finite amplitude described by .in this paper , we have presented a monte - carlo method to investigate a phase space described by a deterministic but irregular tree ( i.e. the number of branches at each node is not fixed ) . with a nave random climbing on the tree ( the _ one_-squirrel method ) ,the probability of a path depends on the number of branches encounter at each node . for the meanders problem ,the ratio between extremal weights increases exponentially with the size : consequently the most part of the computer time is devoted to generate configurations with small weight , and only a exponentially small number of configurations with high weight contribute efficiently to the average .  with the _ multi_-squirrel method ,the distribution becomes almost flat : the bias , i.e. the ratio between extremal weights increases very slowly and never exceeds 3 in our simulations .moreover , this bias is exactly known during the simulation , then it can be corrected to average over all meanders with a uniform distribution . as usual with monte - carlo simulations , results suffer from statistical fluctuations which decrease , in the best case , like the square root of the computer time .  after a simulation on a parallel computer with 3 years of cpu time in single processor units , results with small errors barshave been obtained for meanders up to size . under some hypothesis inspired by the analogy with random walks problems ,large extrapolation can be done for the enumeration ( see eq . [ b1 ] and fig . [ lnr ] ) , for the distribution ( see eq .[ vtheta ] and fig .[ pnw ] ) and the average of the winding number ( see eq .[ vnu ] and fig .[ fnu ] ) and the shape ( see section [ haa ] ) of meanders .  from a monte - carlo point of view , the proposed algorithm can be used , in principal , for any combinatorial problem described by a tree : the essential ingredient is to know , for a given node , the number of branches .however algorithms are rare in this kind of problems and better variants or other algorithms could be without doubt invented .  about the meanders , with these estimates , it appears that the critical exponent are not simple fractions as or , as conjectured by previous studies  .of course , monte - carlo simulations can not determine the exact values , but can confirm or invalidate analytical proposals , while waiting for a rigorous solution .   j. des cloizeaux , g. jannink , _ les polymres en solution : leur modlisation et leur structure _ , ( les editions de physique , 1987 ) ; _ polymers in solution , their modelling and structure , _( clarendon press , 1990 ) .p. di francesco , o. golinelli , e. guitter , _ meanders _ , the mathematical beauty of physics : in memory of claude itzykson : saclay , 5 - 7 june , 1996 ; advanced series in mathematical physics : vol .drouffe and zuber , world scientific ) .

Abstract: we study the statistics of meanders , i.e. configurations of a road crossing a river through bridges , and possibly winding around the source , as a toy model for compact folding of polymers . we introduce a monte - carlo method which allows us to simulate large meanders up to . by performing large extrapolations , we give asymptotic estimates of the connectivity per bridge , the configuration exponent , the winding exponent and other quantities describing the shape of meanders .   keywords : folding , meanders , monte - carlo , tree


this paper is motivated by the need for a robust implementation of solid sweeps in solid modeling kernels .the solid sweep is of course , the envelope surface of a solid which is swept in space by a family of rotations and translations .the uses of sweeps are many , e.g. , in the design of scrolls  , in cnc machining verification  , to detect collisions , and so on .see appendix for an application of solid sweep in designing scrolls , where we describe a modeling attempt using an existing kernel and its limitations .constant radius blends can be considered as the partial envelope of a sphere moving along a specified path . as with blends, it is expected that a deeper mathematical understanding of solid sweep will lead to its rapid deployment and use .a robust implementation of solid sweep poses the following requirements : ( i ) allow for input models specified in the industry - standard brep format , ( ii ) output the sweep envelope in the brep format , with effective evaluators , and finally , ( iii ) perform body - check , i.e. , a check on the orientability , non - self - intersection , detection of singularities and so on .thus there are some `` local '' parts and some `` global '' parts to the problem .it is generally recognized that the harder parts of the local theory is in the smooth case , i.e. , when faces meet each other smoothly . for in the non - smooth case , the added complexity in the local geometry of the sweep is exactly that of a curve moving in 3-space .this is of course well understood , and offered by many kernels as a basic surface type . as far as we know ,the global situation in the non - smooth case , i.e. , the topological structure of edges and vertices ( i.e. , the 1-cage ) of the sweep has not been elucidated , but is also generally assumed to be simpler than the smooth case .in fact , much of existing literature has focused on a smooth single - face solid , as the key problem  .  in this paper , we focus on the smooth multi - face solid . in section [ simplesec ] , we start with the mathematical structure of the simple sweep ( i.e. , one without singularities and self - intersections ) . by the calculus of curves of contact, we set up a correspondence between the faces , edges and vertices of the envelope with those of the swept solid .this sets up the brep structure of the envelope .next , we define the funnel as the parametrization space of a face of the envelope and construct a parametrization .we further elucidate the structure of the bounding edges / vertices of a face and provide several examples of simple sweeps from a pilot implementation .  in section [ simplesisec ], we examine the trim structures .the funnel of section [ simplesec ] will remain the ambient parametrization of the faces .the correspondence will help us define the trim areas and trim curves which must be excised to form the correct envelope .we then define the function and use it to define elementary and singular trim curves .  in section [ decompsec ], we start with the decomposable sweep , i.e. , one which may be partitioned into a suitable small collection of simple sweeps .the final envelope is obtained by stable ( transversal ) boolean operations on this collection .we show that the trim curves so obtained are elementary .we next define an invariant on the funnels , which is robustly and efficiently computable and we show that on ( all ) the funnels characterizes decomposability .this is an important step in the robust implementation of sweeps .  in section [ thetasec ] , we prove some of the properties of such as its invariance and show that it is the determinant of the transformation connecting two 2-frames on the envelope , and is thus an easily computable function on the surface .we show that the curve on the funnel is also the singular locus for the envelope surface . via a geometric theorem, we also show that the function matches the one by  for implicitly defined surfaces and using the so - called inverse trajectory .  in section [ nondecompsec ], we define the singular trim curve , i.e. , where may hit zero .we show that there is a correspondence between singular trim curves and the curves in the zero - locus of .we also show that ( i ) singular trim curves make contact with the curves , and ( ii ) excision at the singular trim curves removes all singularities of the envelope except at these points of contact . furthermore , these points are easily and robustly computed .in section [ conclusionsec ] we summarize what has been achieved , viz ., that the decomposability and the zero - locus of complement to give a complete understanding of all trim curves .we also discuss some implementation issues and extensions .* previous work *  we now review existing related work .perhaps the most elaborate proposal for the sweep surface is the sweep envelope differential equations  approach , where the authors ( i ) assume that surface being swept is implicitly given by a function , and ( ii ) derive a differential equation whose solution is the envelope . for any point on the initial curve of contact , a runge - kutta marchingyields a trajectory such that ( i ) , and ( ii ) , the curve of contact at time .these trajectories presumably serve as the iso - parametric lines . determining whether is in the trimming set solved by using the inverse trajectory condition .this is implemented by using the second derivative of the function , where is the inverse trajectory of point .  on the global front, the building of the envelope is done by selecting a collection of points on the initial curve of contact , developing trajectories , testing for membership in and then using the points which pass to construct an approximation to the envelope .the drawbacks are clear . typically , constructing an which defines is difficult .furthermore , the choice of seems to determine many computational and parametric issues , which is undesirable .the inverse - trajectory check remains poorly conditioned , especially when the second derivative of the function w.r.t . is zero .the structure of the envelope is unknown where this derivative is zero .a global understanding of and the nature of the trim curves is missing .  in  ,while classifying points for sweeping solids , the authors give a membership test for a point in the object space to belong inside , outside or on the boundary of the swept volume by using inverse trajectory of that point .a curve - solid intersection is required to be computed for each point membership query which is computationally expensive , especially when the intersection is non - transversal , as noted by the authors themselves .such high degree of computational complexity is prohibitive for a practical implementation .  in the authors work with 2d shapes and 2d motions and quantify singularities using inverse trajectories .this work is based on the computational framework described in  and involves computing intersections between 2d curves and 2d shapes .the authors remark that this work can be extended to the 3-dimensional case involving intersections between 3d curves and 3d solids .this approach has the same drawback as  , namely a high computational cost .  in trimming self - intersections in swept volumes , the authors detect self - intersections by computing approximate curves of contact at a few discrete time instances which are then checked for intersections .approximations are introduced at multiple levels , hence an accurate solution can not be expected from this method .in this section we formulate the boundary of the volume obtained by sweeping a solid along a given trajectory .we will use the boundary representation , also known as brep , which is a popular standard for representing a compact and oriented solid by its boundary .the boundary separates the interior of from the exterior of and is represented using a set of _ faces _ , _ edges _ and _ vertices_. see figure [ conefig ] for the brep of a solid where different faces are colored differently .faces meet in edges and edges meet in vertices .the brep consists of two interconnected pieces of information , viz . , the geometric and the topological .the geometric information consists of the parametric description of the faces and edges while the topological information consists of orientation of the geometric entities and adjacency relations between them .  in this paperwe consider solids whose boundary is formed by faces meeting smoothly . in the case when the faces do not meet smoothly , the added complexity in the local geometry of the sweep is exactly that of a curve moving in 3-space .this is of course well understood , and offered by many kernels as a basic surface type .the global geometry and topology for this case will be described in a later paper .[ trajectorydef ] a * trajectory * in is specified by a map where is a closed interval of , .the parameter represents time .we assume that is of class for some , i.e. , partial derivatives of order up to exist and are continuous .we make the following key assumption about .[ genericassum ] the tuple is in a _general position_.  [ envldef ] the * action * of ( at time in ) on is given by .the * swept volume * is the union and the * envelope * is defined as the boundary of the swept volume .clearly , for each point of there must be an and a such that .this sets up the following correspondence relation .[ corrdef ] the * correspondence * is the set of tuples for , we set .similarly , for , we define .we will denote the interior of a set by .it is clear that .therefore , we have  [ intlem ] if , then for all , .    thus , the points in interior of do not contribute to at all and .this sets up the brep structure for . in the sweep example shown in figure [ conefig ] , the correspondence is illustrated via color coding , i.e. , for , the points and are shown in the same color . the general position assumption on be formulated as the condition that the induced brep topology of remains invariant under a small perturbation of .[ preimagelem ] assuming general position of , for any , there are at most three distinct tuples for which belong to ._ for distinct tuples , it is clear that , for otherwise . therefore and intersect at point . by assumption [ genericassum ]this intersection is transversal .further , by the same assumption , at most surfaces may intersect in a point .  [ trajxdef ] for a point , define the * trajectory of * as the map given by and the velocity as .for a point , let be the unit outward normal to at .define the function as thus , is the dot product of the velocity vector with the unit normal at the point .proposition [ glem ] gives a necessary condition for a point to contribute a point on at time , namely , , and is a rewording in our notation of the statement in  that _ the candidate set is the union of the ingress , the egress and the grazing set of points_.  [ glem ] for and  ] , we define the following subsets of the parameter space   the set will be referred to as the * funnel*.    by assumption [ genericassum ] about the general position of it follows that for all , the gradient^t \neq \bar{0} ] , either ( i ) and , or ( ii ) and , or ( iii ) ._ this follows from prop .[ glem ] and definition [ fedef ] . figure [ wireframefig ] shows the edges of the envelope for the sweep example shown in figure [ conefig ] .the correspondence for one of the edges of the envelope is also marked .let denote the funnel corresponding to the contact set generated by face .the edge in parameter space which bounds is given by which we will denote by .note that is smooth if at all points in .a vertex on will , in general , give rise to a set of vertices on .we further restrict the correspondence to as . as is smooth , there is a well - defined normal at .hence we may define the function as .if is on the boundary of a face , will have a set of coordinates in the parameter space of the surface the face , say , so that .it is easy to see that if and  ] . in other words , \in l(p_0) ] , i.e. , for all  ] , i.e. , and .we have shown that for , iff contains an interval .hence , is discrete iff .  as , by lemma [ preimagelem ], it follows that at all but finitely many points , is of cardinality and at remaining points it is of cardinality .  we classify trim curves as follows .  [ trimcclassifydef ] a curve of is said to be * elementary * if there exists such that for all , .it is said to be * singular * if .figures [ ptrimcurveafig](a ) and [ ptrimcurveafig](b ) schematically illustrate elementary and singular p - trim curves on respectively .further observe that , in case ( a ) and 0 in case ( b ) .  before proceeding further, we introduce the following notation : for , .[ transinterlem ] all but finitely many points of elementary trim curves lie on the transversal intersections of two surface patches and the remaining points lie on the transversal intersection of three surface patches where , for , are subintervals ._ suppose that all curves of are elementary , i.e. , such that for all , .by lemma [ trimclem ] , all but finitely many points have two points and in such that .let ) ]. then . from section [ geomthetasec ]we know that and are tangential to and respectively at .by assumption [ genericassum ] about general position of , and intersect transversally at .hence , and intersect transversally at .  at most finitelymany points have three points and in such that . by an argument similar to above, it can be shown that lies on the transversal intersection of three surface patches for corresponding to appropriate subintervals .  figure [ toolfig ] shows an example in which a capsule is swept along a helical path while rotating about -axis .the trim curves are elementary .-axis.(b ) the contact set restricted to interval  ] with the trim set excised.,title="fig : " ]we now consider sweeps , which though not simple , can be divided into simple sweeps by partitioning the sweep interval so that the trim curves can be obtained by transversal intersections of the contact sets of the resulting simple sweeps . given an interval , we call a partition of into consecutive intervals to be of width if .[ decompdef ] we say that the sweep is * decomposable * if there exists such that for all partitions of of width , each sweep is simple for .a sweep which is not decomposable is called * non - decomposable*.  figure [ decompnondecompfig ] schematically illustrates the difference between decomposable and non - decomposable sweeps .the example shown in figure [ toolfig ] is of a decomposable sweep in which partitioning the sweep interval into 2 equal halves will result in 2 simple sweeps .[ decomplem ] the sweep is decomposable iff .further , if then all the p - trim curves are elementary ._ suppose first that .let be a partition of of width .we show that is simple for .let and be the envelope and the contact set for respectively . by proposition [ glem ] , ( modulo end - caps ) ,it needs to be shown that .suppose not .let such that for some .then , , i.e. , for some .let for .it follows that , leading to a contradiction .hence , is decomposable .suppose now that is decomposable with width - parameter ( cf . definition [ decompdef ] ) .consider a point and let .let  ] .further , let and be the envelope and contact - set for the sweeps respectively .observe that for .let .as is decomposable with width - parameter , both and are simple , and hence , for .therefore , belongs to and . by lemma[ simplelem ] , and are both singleton sets .further , for .hence , .since for all , , we conclude that .suppose that .since for all if follows that all the p - trim curves are elementary .  the above proposition provides a _ natural _test for decomposability .further , coupled with lemma [ transinterlem ] , for a decomposable sweep , the problem of excising the trim set can be reduced to transversal intersections .however , note that , the very definition of is _ post - facto _ as it relies on the trim structures . besides , it is the infimum value of the not necessarily continuous function and is difficult to compute .thus , the above test of decomposability is not effective .one of the key contributions of this paper is a novel geometric ` invariant ' function on the funnel which is computed in closed form and serves the following objectives .1 .  quick / efficient and simple detection of decomposability of sweeps , which occur most often in practice .2 .  generation of trim curves for non - decomposable sweeps .3 .  quantification and detection of singularities on the envelope .  for a point ,let .recall from subsection [ paramsec ] that ,  ] swept along a curvilinear trajectory given by , ,  ] and ( ii ) ,  ] .  [ invtrajdef ] given a trajectory , the * inverse trajectory * is defined as the map given by .thus , for a fixed point , the inverse trajectory of is the map given by .the range of is .we list some of the facts about in the appendix which will be used in proving theorem [ lambdalem ] .  for the inverse trajectory of a point , let be the projection of on .let be the signed distance of from .if the point is in , ( the exterior of ) or on the surface , then is negative , positive or zero respectively .then we have , where is the projection of on along the unit outward pointing normal to at .this is illustrated in figure [ type2lsifig ] .thus the following relation holds for .   intersects . ][ lambdalem ] for , where is the normal curvature of at along velocity , is the unit outward normal to at and .see appendix for the proof .  from theorem [ lambdalem ]it is clear that the function is intimately connected with the curvature of the solid and that of the trajectory .it is easy to see that the function is identical to the function defined in  for implicitly defined solids , albeit , is invariant of the function defining the solid as well as the parametrization of the same ._ suppose that for all , .for , let denote the -coordinate of .consider the set of points and .by the general position assumption , is a collection of smooth curves in . for , let denote the unique point in such that and .further , we define . let .consider two cases as follows :  * case ( i ) * : , i.e. , there exists a sequence in a curve of such that .hence there exists ( closure of ) which is a limit point of .since and is free from self - intersections , we have that .hence , for a small neighborhood of in , we may parametrize the smooth curve by a map so that and , for , and .let .note that .now ,   hence , since , the map fails to be an immersion at and by lemma [ singlem ] we get that , which is a contradiction to the hypothesis .* case ( ii ) * : .let be a partition of of width .let and denote the funnel and the contact set corresponding to subinterval .then it is clear that for each , is a diffeomorphism , i.e. , for each , for all , .we show that the subproblems are simple for all .suppose not , i.e. , for some , there exists such that for some .hence the trim set is not empty . by lemma [ transinterlem ] ,for all but finitely many points in there are two points such that .if and then it follows that leading to contradiction .hence , the subproblems are simple for all .it follows that is decomposable with width - parameter .hence we have proved that if for all , then the sweep is decomposable .suppose now that there exists such that .let .recall the definition of the function from equation [ lambdaeq ] and relation from theorem [ lambdalem ] .clearly , if , then is a local maxima of the function and the inverse trajectory of intersects .so , there exists such that for all , there exists such that .hence , the interval \subset l(p) ] , either ( i ) and , or ( ii ) and or ( iii ) ._ define and as  and respectively .we define the following objects in where the fourth dimension is time .let and .note that is a four dimensional topological manifold and is a three dimensional submanifold of .further , a point lies in if and .further , if  ] , .we now note a few useful facts about .we assume without loss of generality that and . denoting the derivative with respect to by , we have since we have , differentiating eq .[ aso3eq ] w.r.t . we get using eq .[ ybardoteq ] and eq .[ adottnoteq ] we get differentiating eq .[ ybardoteq ] w.r.t .time we get using equations [ ybardotdoteq ] , [ adottnoteq ] and [ adotdottnoteq ] we get proof . _ recall the definition of function as differentiating eq . [ lambdaeq ] with respect to time and denoting derivative w.r.t . by , we get at , . since , it follows from eq .[ ybardottnoteq ] that .it is easy to verify that .hence , from eq .[ ddotlambdaeq ] and eq .[ ybardotdottnoteq ] it follows that since for all in some neighbourhood of , we have that . hence . hence = . here is the differential of the gauss map , i.e. the curvature tensor of at point . using this in eq .[ ddotlambdatnoteq ] and the fact that , we get recalling that  here and where is the shape operator ( differential of the gauss map ) of at . also , and .assume without loss of generality that and , hence , and .using eq .[ adottnoteq ] and the fact that we get from eqs .[ lsi2eq ] and [ lsirelationeq ] and the fact that we get . now describe the parametrization of assuming that the sweep is simple .we obtain a procedural parametrization of which is an abstract way of defining curves and surfaces .this approach relies on the fact that from the user s point of view , a parametric surface(curve ) in is a map from ( ) to and hence is merely a set of programs which allow the user to query the key attributes of the surface(curve ) , e.g. its domain and to evaluate the surface(curve ) and its derivatives at the given parameter value .this approach to defining geometry is especially useful when closed form formulae are not available for the parametrization map and one must resort to iterative numerical methods .we use the newton - raphson(nr ) method for this purpose . as an example , the parametrization of the intersection curve of two surfaces is computed procedurally in  .this approach has the advantage of being computationally efficient as well as accurate . for a detailed discussion on the procedural framework ,see  .the computational framework is as follows . given and , an approximate funnel is first computed , which we will refer to as the seed surface .now , when the user wishes to evaluate or its derivative at some parameter value , a nr method will be started with seed obtained from the seed surface .the nr method will converge , upto the required tolerance , to the required point on , or to its derivative , as required . here , the precision of the evaluation is only restricted by the finite precision of the computer and hence is accurate .it has the advantage that if a tighter degree of tolerance is required while evaluation of the surface or its derivative , the seed surface does not need to be recomputed .thus , for the procedural definition of we need the following :  1 .  an nr formulation for computing points on and its derivatives , which we describe in section [ nrformsubsec ] 2 .  seed surface for seeding the nr procedure , which we describe in section [ seedsubsec ]  recall that by the non - degeneracy assumption , is the union of .this suggests a natural parametrization of in which one of the surface parameters is time .we will call the other parameter and denote the seed surface by which is a map from the parameter space of to the parameter space of , i.e. and while the point may not belong to , it is close to . in other words, is close to .we call the image of the seed surface through the sweep map as the approximate envelope and denote it by , i.e. .we make the following assumption about .note that this is not a very strong assumption and holds true in practice even with rather sparse sampling of points for the seed surface .we now describe the newton - raphson formulation for evaluating points on and its derivatives at a given parameter value .recall that the points on were characterized by the tangency condition . introducing the parameters of , we rewrite this equation : so , given , we have one equation in two unknowns , viz . and . is defined as the intersection of the plane normal to the iso-(for ) curve of at with the iso-(for ) curve of which is nothing but .recall that is given by where obey eq .[ envlcondpareq ] .henceforth , we will suppress the notation that and are functions of and .also , all the evaluations will be understood to be done at parameter values .the tangent to iso- curve of at is given by hence , is the solution of simultaneous system of equations [ envlcondpareq ] and [ planeorthoeq ] eq .[ envlcondpareq ] and eq .[ planeorthoeq ] give us a system of two equations in two unknowns , and and hence can be put into nr framework by computing their first order derivatives w.r.t and . for any given parameter value , we seed the nr method with the point and solve eq .[ envlcondpareq ] and eq .[ planeorthoeq ] for and compute .  having computed now compute first order derivatives of assuming that they exist . in order to compute , we differentiate eq .[ envlcondpareq ] and eq .[ planeorthoeq ] w.r.t . to obtain eq .[ derpeq1 ] and eq .[ derpeq2 ] give a system of two equations in two unknowns , viz ., and and can be put into nr framework by computing first order derivatives w.r.t . and .note that eq .[ derpeq1 ] and eq .[ derpeq2 ] also involve and whose computation we have already described .after computing and , can be computed as . can similarly be computed by differentiating eq .[ envlcondpareq ] and eq .[ planeorthoeq ] w.r.t .higher order derivatives can be computed in a similar manner .the seed surface is constructed by sampling a few points on the funnel and fitting a tensor product b - spline surface through these points .for this , we first sample a few time instants , say , from the time interval of the sweep . for each , we sample a few points on the pcurve of contact . for this, we begin with one point on and compute the tangent to at , call it . then is used as a seed in newton - raphson method to obtain the next point on and this process is repeated .

Abstract: in this work , we propose a detailed computational framework for modelling the envelope of the swept volume , that is the boundary of the volume obtained by sweeping an input solid along a trajectory of rigid motions . our framework is adapted to the well - established industry - standard brep format to enable its implementation in modern cad systems . this is achieved via a `` local analysis '' , which covers parametrization and singularities , as well as a `` global theory '' which tackles face - boundaries , self - intersections and trim curves . central to the local analysis is the `` funnel '' which serves as a natural parameter space for the basic surfaces constituting the sweep . the trimming problem is reduced to the problem of surface - surface intersections of these basic surfaces . based on the complexity of these intersections , we introduce a novel classification of sweeps as either decomposable or non - decomposable . further , we construct an _ invariant _ function on the funnel which efficiently separates decomposable and non - decomposable sweeps . through a geometric theorem we also show intimate connections between , local curvatures and the inverse trajectory used in earlier works as an approach towards trimming . in contrast to the inverse trajectory approach , is robust and is the key to a complete structural understanding , and an efficient computation of both , the singular locus and the trim curves , which are central to a stable implementation . several illustrative outputs of a pilot implementation are included .   sweeping , boundary representation , parametric curves and surfaces


computation of many interior eigenvalues and eigenvectors of a large sparse symmetric ( or hermitian ) matrix is a frequent task required for electronic structure calculations in quantum chemistry , material science , and physics .starting from the time - independent schrdinger equation , where is the hamilton operator of electrons in the material under consideration , one is interested in all solutions of this equation with energies within a small window inside of the spectrum of .interior eigenvalues are required because electrical or optical material properties are largely determined by the electronic states in the vicinity of the fermi energy , which separates the occupied from the unoccupied electron states .the width of the energy window is given by , e.g. , the temperature or the magnitude of an applied electric voltage .recently , the computation of interior eigenvalues has gained new thrust from the fabrication of graphene  and topological insulator  materials , which exhibit unconventional electron states that differ from those observed in traditional semiconductors or metals . in graphene ,electrons at the fermi level follow a linear dispersion reminiscent of relativistic dirac fermions , while the topologically protected surface states of topological insulators are natural candidates for quantum computing and quantum information storage . because the properties of graphene and topological insulator devices depend directly on such electron states close to the center of the spectrum , their modeling and design requires the solution of ever larger interior eigenvalue problems  .motivated by the rapidly developing demands on the application side we introduce in this paper a high - performance implementation of chebyshev filter diagonalization ( chebfd ) , a straightforward scheme for interior eigenvalue computations based on polynomial filter functions .chebfd has a lot in common with the kernel polynomial method ( kpm )  and is equally well suited for large - scale computations  .our goal is to show that a well - implemented chebfd algorithm can be turned into a powerful tool for the computation of many interior eigenvalues of very large sparse matrices .here we include results for the computation of central eigenvalues of a -dimensional topological insulator matrix on nodes of the federal supercomputer system `` supermuc '' at lrz garching , at tflop / s sustained performance .chebfd is , to our knowledge , presently the only approach that can solve interior eigenvalue problems at this scale .while the present paper focuses on high - performance applications of chebfd , and less on algorithmic improvements , we will discuss its theoretical foundations in considerable detail .we will analyze especially how the algorithmic efficiency of polynomial filtering depends on the number of search vectors in the computation , and not only on the quality of the filter polynomials .using a larger search space and lower filter polynomial degree can improve convergence significantly .conceptually , chebfd is a massively parallel block algorithm with corresponding demands on computational resources , which makes performance engineering mandatory .one should note that the use of chebyshev filter polynomials has a long tradition in the solution of linear systems  , as a restarting technique for ( krylov ) subspace iterations  , and for the computation of extreme and interior eigenvalues in quantum chemistry applications  .the latter methods can be divided roughly into two classes : methods which are based explicitly or implicitly on time propagation of a wave function with chebyshev techniques and ( re-)construction of eigenstates from fourier inversion of the computed time signal ( as , e.g. , in ref . ) , and methods that use chebyshev filters for iterated subspace projection steps ( see , e.g. , ref . for a recent contribution ) .chebfd belongs to the latter class of methods .  the rationale behind the use of polynomial filters in chebfd instead of rational filter functions ( as , e.g. , in feast  or the cirr method  ) or of spectral transformations of the eigenvalue problem  is that sparse matrix factorization is not feasible for the very large matrices in our target applications .this prevents the use of direct solvers such as pardiso  or ilupack  , which have been used successfully for , e.g. , anderson localization  . in some applicationsthe matrix is not even stored explicitly but constructed ` on - the - fly '  .therefore , we assume that the only feasible sparse matrix operation in chebfd is sparse matrix vector multiplication ( spmvm ) .  if a matrix is accessed exclusively through spmvms the only expressions that can be evaluated effectively are polynomials \vec v ] , with half width , is contained within the search interval  ] that encloses the spectrum of the matrix under consideration . typically , it is and the target interval center lies far away from , .the symmetric situation is especially challenging ., scaledwidth=50.0% ]  the eigenvalue problem described in the introduction can be specified as follows : given a symmetric ( or hermitian ) matrix , compute the eigenpairs within a prescribed target interval  ] is the exact projection onto the target space , that is , \vec v_i = \vec v_i ] for .the eigenpairs in the target interval can then be obtained as follows : start with sufficiently many ( random ) search vectors , for with , and compute the filtered vectors \vec x_k ] that contains the spectrum of ( see fig . [fig : sketch ] ) . here , is the -th chebyshev polynomial of the first kind , with recurrence notice the scaling of the argument  ] .the expansion coefficients are given by   \frac2 { \pi n } \big ( \sin \big(n \arccos ( \alpha { \overline\lambda}+ \beta ) \big ) - \sin \big ( n \arccos ( \alpha { \underline\lambda}+ \beta ) \big )  \big )  & \text { if } n \ge 1 \;.   \end{cases}\end{aligned}\ ] ] here , the first expression results from orthogonality of the with respect to the weight function , the second expression is obtained by evaluating the integral for the rectangular function .truncation of the infinite series  leads to a polynomial approximation to , which , however , suffers from gibbs oscillations ( see fig . [fig : filters ] ) .the solution to the gibbs phenomenon exploited in , e.g. , kpm is to include suitable kernel damping factors in the expansion .this gives the polynomial approximation of degree .three examples for the kernel coefficients are given in table [ tab : kernel ] , including the jackson kernel  known from kpm  .the resulting approximations to are plotted in fig .[ fig : filters ] . in chebfdwe will use the lanczos kernel with , which tends to give faster convergence than the other choices ..coefficients of the fejr , jackson , and lanczos kernel ( cf . ref .the lanczos kernel has an additional parameter ( later we will use ) . [cols="^,^,^ " , ]   the good scalability and high performance of our chebfd implementation allows us to deal with the very large matrices that arise in the theoretical modeling of realistic systems . to illustrate the physics that becomes accessible through large - scale interior eigenvalue computations we consider the example of gate - defined quantum dots in a topological insulator ( fig .[ fig : dot1 ] ) , a primary example for the functionalization of topological materials .  in this example a finite gate potential is tuned to a value ( ) such that the wave functions of the innermost eigenstates , i.e. , those located at the fermi energy , are concentrated in the circular gate regions .minimal changes of the gate potential can lead to large changes of the wave functions , which gives rise to a transistor effect that allows for the rapid switching between an ` on ' and an ` off ' state .while this effect can be studied already for small systems , or is even accessible to analytical treatment due to the high symmetry of the situation , the modeling of realistic systems requires the incorporation of disorder at the microscopic level , which results either from material imperfections or deliberate doping with impurity atoms .the natural question then is how disorder modifies the transistor effect . to answer this question the interior eigenstatesmust now be computed numerically , and large system sizes are required to correctly describe the effect of the microscopic disorder on the mesoscopic quantum dot wave functions .an example for such a computation with chebfd is shown in fig .[ fig : dot2 ] . eigenstates close to the fermi energy are still concentrated in the gate regions , although the circular shapes visible in the wave function from fig .[ fig : dot1 ] are washed out by disorder .interestingly , the two nearby eigenstates shown in the picture are concentrated in different gate regions , which is possible because disorder breaks the perfect square symmetry of the arrangement .therefore , disorder can induce a new transistor effect , where a small gate potential switches an electric current between different sides of the sample .full exploration of this effect certainly requires more realistic modeling of the situation .as the present examples demonstrate , chebfd can become a relevant computational tool for such explorations .  , i.e. , the matrix dimension is , with disorder .right panel : wave function in the plane belonging to the two eigenvalues marked by green circles in the left panel.,title="fig:",scaledwidth=80.0% ] + , i.e. , the matrix dimension is , with disorder .right panel : wave function in the plane belonging to the two eigenvalues marked by green circles in the left panel.,title="fig:",scaledwidth=80.0% ]polynomial filtering techniques such as the chebfd scheme examined in the present paper are powerful tools for large - scale computations of many interior eigenvalues of sparse symmetric matrices .these techniques combine two aspects . on the one hand, they use polynomial filter functions to damp out eigenvector contributions outside of the target interval .polynomial filters allow for simple evaluation through spmvm , and are thus applicable to large matrices .on the other hand , these techniques rely on large search spaces , which can lead to fast convergence also for moderate quality of the polynomial filters .chebfd is a straightforward implementation of polynomial filtering . despite its algorithmic simplicity , chebfd is well suited for large - scale computations , such as the determination of the central eigenpairs of a -dimensional matrix presented here . to our knowledgepresently no other method is capable of computations at this scale .these promising results indicate that chebfd can become a viable tool for electronic structure calculations in quantum chemistry and physics , and for research into modern topological materials .the present chebfd scheme can be refined along several directions .first , the construction of better polynomial filters could improve convergence .second , adaptive schemes that adjust the polynomial degree and the number of search vectors according to the dos , which can be computed with increasing accuracy in the course of the iteration , might prove useful .third , ritz values should be replaced by harmonic ritz values in the computation , together with an improved convergence criterion and locking of converged eigenpairs .note , however , that our theoretical analysis of chebfd shows that none of these improvements is truly crucial for the ( fast ) convergence of polynomial filtering : the most important aspect is the `` overpopulation '' of the target space with search vectors , i.e. , the use of a sufficiently large search space .a more radical alternative to chebfd is the replacement of polynomial filters by rational functions as , e.g. , in the feast and cirr methods . from the theoretical point of view, rational functions always lead to better convergence. however , evaluation of rational functions of sparse matrices is non - trivial .standard iterative solvers require too many additional spmvms to be competitive with polynomial filters , unless they can exploit additional structural or spectral properties of the matrix  . at presentit is not clear whether non - polynomial filter functions can succeed in large - scale problems of the size reported here .this question is currently under investigation .the chebfd algorithm implemented and investigated in the present work should be of particular interest to physicists and chemists who increasingly require high - performance tools for the computation of interior eigenvalues in present and future research applications .the library , the chebfd code , and generator routines for all test cases are publicly available from the essex repository .the research reported here was funded by deutsche forschungsgemeinschaft through priority programmes 1459 `` graphene '' and 1648 `` software for exascale computing '' .the authors gratefully acknowledge the gauss centre for supercomputing e.v .( http://www.gauss-centre.eu ) for funding this work by providing computing time on the gcs supercomputer supermuc at leibniz supercomputing centre ( lrz , http://www.lrz.de ) through project pr84pi .the -based implementation of chebfd will become available as part of the sparse solver library prepared in the essex project .10 url # 1`#1`urlprefixhref # 1#2#2 # 1#1       m. kreutzer , g. hager , g. wellein , a. pieper , a. alvermann , h. fehske , performance engineering of the kernel polynomial method on large - scale cpu - gpu systems , in : proc . of the 29th ieee international parallel & distributed processing symposium ipdps15 , 2015 , pp. 417426 .m. galgon , l. krmer , b. lang , a. alvermann , h. fehske , a. pieper , g. hager , m. kreutzer , f. shahzad , g. wellein , a. basermann , m. rhrig - zllner , j. thies , improved coefficients for polynomial filtering in essex , preprint ( 2016 ) .m. kreutzer , j. thies , m. rhrig - zllner , a. pieper , f. shahzad , m. galgon , a. basermann , h. fehske , g. hager , g. wellein , ghost : building blocks for high performance sparse linear algebra on heterogeneous systems , arxiv:1507.08101 ( preprint ) ( 2015 ) .m. rhrig - zllner , j. thies , m. kreutzer , a. alvermann , a. pieper , a. basermann , g. hager , g. wellein , h. fehske , increasing the performance of the jacobi - davidson method by blocking , siam j. sci .37 ( 2015 ) c697c722 .j. treibig , g. hager , g. wellein , likwid : a lightweight performance - oriented tool suite for x86 multicore environments , in : proceedings of psti2010 , the first international workshop on parallel software tools and tool infrastructures , san diego ca , 2010 .m. galgon , l. krmer , j. thies , a. basermann , b. lang , on the parallel iterative solution of linear systems arising in the feast algorithm for computing inner eigenvalues , parallel comput .49 ( 2015 ) 153163 .

Abstract: we study chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices . in this technique the subspace projection onto the target space of wanted eigenvectors is approximated with filter polynomials obtained from chebyshev expansions of window functions . after the discussion of the conceptual foundations of chebyshev filter diagonalization we analyze the impact of the choice of the damping kernel , search space size , and filter polynomial degree on the computational accuracy and effort , before we describe the necessary steps towards a parallel high - performance implementation . because chebyshev filter diagonalization avoids the need for matrix inversion it can deal with matrices and problem sizes that are presently not accessible with rational function methods based on direct or iterative linear solvers . to demonstrate the potential of chebyshev filter diagonalization for large - scale problems of this kind we include as an example the computation of the innermost eigenpairs of a topological insulator matrix with dimension derived from quantum physics applications .   interior eigenvalues , chebyshev filter polynomials , performance engineering , quantum physics , topological materials