a generic model for long self avoiding chain molecules a generic model which applies rigorously to finite chains in the absence of inter segmental interactions and to subchains of infinite chains with practically arbitrary interactions furnishes a simple recurrence relation self avoiding walks on the diamond lattice saw d form the paradigmatic example for polymer science the generic model as a whole is solved in terms of parameters for configurational statistics asymptotically large number n of segments and on stated plausible conjectures by classical methods i e without postulating a singularity of power law form the appropriate generalisation of the power law form of scaling theory emerges very simply e g in terms of kummer s hypergeometric function extensive computations on the saw d model lead to two correction terms to the crude scaling form in this way some arguments in the literature on asymptotic theories can be settled several examples illustrate the danger of mistaking non asymptotic experimental results for those desired in the asymptotic range thus contrary to his own conclusions and those of fleming 1979 brun s data 1977 data for freely hinged hard sphere chains are here reconciled with the universal exponent gamma of le guillou and zinn justin 1977 for the saw d model subject always to further refinements which will never produce a final answer the present experimental estimate for gamma is also shown to be about 1 2 in line with conjectural estimates from many forms of modern theory

spiralling self avoiding walks an exact solution an exact solution is presented to a problem of spiralling self avoiding walks on the square lattice recently proposed by privman 1983 for n to infinity the number of n step spiral walks increases as cn approximately 2 23 5 4 pi n 7 4 exp 2 pi n 3 1 2 and their root mean square end to end distance behaves as rn approximately 1 2 square root pi 1n1 2 log n

on the number of spiral self avoiding walks on a triangular lattice an elementary derivation of the complete asymptotic expansion for the number sn of spiral self avoiding walks with n steps on a triangular lattice is given

spiral self avoiding walks on a triangular lattice end to end distance an exact solution is obtained for the root mean square end to end distance rn of spiral self avoiding walks with n steps on a triangular lattice for n to infinity rn increases as 2 pi 1 6n 1 2 log n

spiral self avoiding walks on a triangular lattice an exact result is obtained for the number sn of spiral self avoiding walks with n steps on a triangular lattice for n to infinity sn increases as 214 3 7 4 pi n 5 4 exp pi 2n 3 1 2

3 d attitude representation of human joints a standardization proposal abstract in view of the singularities asymmetries and other adverse properties of existing three dimensional definitions for joint and segment angles the present paper proposes a new convention for unambiguous and easily interpretable 3 d joint angles based on the concept of the attitude vector as derived from euler s theorem the suggested standard can be easily explained to non mathematically trained clinicians is readily implemented in software and can be simply related to classical cardanic eulerian angles for planar rotations about a coordinate system s axes the proposed convention coincides with the cardanic convention the attitude vector dispenses with the gimbal lock and non orthogonality disadvantages of cardanic eulerian conventions therefore its components have better metrical properties and they are less sensitive to measurement errors and to coordinate system uncertainties than cardanic eulerian angles a sensitivity analysis and a physical interpretation of the proposed standard are given and some experimental results that demonstrate its advantages

on three dimensional spiral anisotropic self avoiding walks two new models of three dimensional anisotropic spiral self avoiding walks are introduced with different types of spiral constraint series expansions for the two models are derived and analysed one model is found to behave like the isotropic three dimensional self avoiding walk while the other model appears to belong to a distinct universality class with exponents nu approximately 0 655 and gamma approximately 1 24 it is argued that for these non markovian undirected unweighted walks the absence of a plane of reflection symmetry in the allowed walks signals a new universality class

spiral growing self avoiding walk a new universality class the author introduces the concept of a spiral growing self avoiding walk sgsaw by imposing privman s spiralling constraint ibid vol 16 p l571 1983 on the growing self avoiding walk introduced by majid et al phys rev lett vol 52 p 1257 1984 and lyklema and kremer ibid vol 17 p l691 1984 he studies the critical properties of the sgsaw using monte carlo simulation the exponent nu associated with the average radius of gyration sn is 0 8 on the other hand the critical exponent nu r associated with the end to end distance rn is much less than nu the latter is in sharp contrast to the result nu nu r for the usual saw directed gsaws are shown to behave like directed usual saws

two dimensional anisotropic spiral self avoiding walks two models of anisotropic spiral self avoiding random walks recently proposed by manna 1984 have been investigated by the method of exact series expansions the number of such n step walks cn appears to behave like cn approximately constant mu nnbeta exp alpha square root n where both mu which is known exactly and the constant factor are model dependent but alpha approximately 0 14 and beta approximately 0 9 appear to be model independent the mean square end to end distance exponent nu 0 855 or 0 02 for both models

effect of anisotropic constraints on self avoiding walks the author studies the effect of four types of anisotropic constraints on self avoiding walks saws on the square lattice in which walks along one lattice direction are constrained differently from those of the other three identically constrained lattice directions by numerical studies the author indicates that all these types of anisotropically constrained saws belong to the universality class of directed self avoiding walks

the spiralling self avoiding walk in a random environment the authors study the spiralling self avoiding lattice walk in a random environment upon scaling the temperature appropriately with system size a phase transition appears in the low temperature phase the walk segments occupy a few low energy positions in the high temperature phase they are effectively free an analogy with the random energy model is pointed out the average size of an n step walk is shown to be asymptotically proportional to n1 2 log n as was known for the homogeneous lattice with a coefficient that increases as the temperature is lowered the spatial distribution of the walk segments is qualitatively different above and below the critical temperature the model also allows for a spin glass interpretation and as such helps to clarify the connection between the concepts of frustration and the chaoticity of the pair correlation both above and below the critical point

characteristics of deterministic and stochastic sandpile models in a rotational sandpile model rotational constraint representing a local external bias generally has a nontrivial effect on the critical behavior of lattice statistical models in equilibrium critical phenomena in order to study the effect of rotational bias in an out of equilibrium situation like self organized criticality a two state quasideterministic rotational sandpile model is developed here imposing rotational constraint on the flow of sand grains an extended set of critical exponents are estimated to characterize the avalanche properties at the nonequilibrium steady state of the model the probability distribution functions are found to obey usual finite size scaling supported by negative time autocorrelation between the toppling waves the model exhibits characteristics of both deterministic and stochastic sandpile models

the finite helical axis of the knee joint a non invasive in vivo study using fast pc mri an understanding of the in vivo knee joint kinematics is critical for the further improvement and validation of knee joint models and for the development of better surgical and rehabilitative protocols unfortunately most studies exploring the finite helical axis fha tend to produce excellent qualitative results but quantitative results are often lacking thus the purpose of this study was to non invasively and in vivo quantify the tibiofemoral fha in a relatively large normal population during volitional knee extension using fast pc mri to report the data relative to consistent coordinate systems making it available for modeling input experimental comparison and for device design to determine the variability of the fha to investigate the screw home mechanism and to test the hypothesis that knee joint kinematics are independent of gender intra and inter subject repeatability was excellent the intra inter subject repeatability of the fha orientation in the frontal and axial planes was 1 8 3 3 and 3 7 6 0 of the average value respectively at the beginning of extension the fha was directed laterally and slightly superiorly and at the end of extension it was directed in the lateral inferior direction indicative of the screw home mechanism the fha location was not fixed during extension there was small but significant differences in all fha parameters between genders and normalizing positional data relative to epicondylar width helped to reduce this difference the data obtained in the current study forms an excellent base for future knee joint modeling and clinical studies

registration of 6 dofs electrogoniometry and ct medical imaging for 3d joint modeling the paper describes a method in which two data collecting systems medical imaging and electrogoniometry are combined to allow the accurate and simultaneous modeling of both the spatial kinematics and the morphological surface of a particular joint the joint of interest joi is attached to a plexiglas jig that includes four metallic markers defining a local reference system r gonio for the kinematics data volumetric data of the joi and the r gonio markers are collected from medical imaging the spatial location and orientation of the markers in the global reference system r ct of the medical imaging environment are obtained by applying object recognition and classification methods on the image dataset segmentation and 3d isosurfacing of the joi are performed to produce a 3d model including two anatomical objects the proximal and distal joi segments after imaging one end of a custom made 3d electrogoniometer is attached to the distal segment of the joi and the other end is placed at the r gonio origin the joi is displaced and the spatial kinematics data is recorded by the goniometer after recording data registration from r gonio to r ct occurred prior to simulation data analysis was performed using both joint coordinate system jcs and instantaneous helical axis iha finally the 3d joint model is simulated in real time using the experimental kinematics data the system is integrated into a computer graphics interface allowing free manipulation of the 3d scene the overall accuracy of the method has been validated with two other kinematics data collection methods including a 3d digitizer and interpolation of the kinematics data from discrete positions obtained from medical imaging validation has been performed on both superior and inferior radio ulna joints i e prono supination motion maximal rms error was 1 degrees and 1 2mm on the helical axis rotation and translation respectively prono supination of the forearm showed a total rotation of 132 degrees for 0 8mm of translation the method reproducibility using jcs parameters was in average 1 degrees maximal deviation 2 degrees for rotation and 1mm maximal deviation 2mm for translation in vitro experiments have been performed on both knee joint and ankle joint averaged jcs parameters for the knee were 109 degrees 17 degrees and 4 degrees for flexion internal rotation and abduction respectively averaged maximal translation values for the knee were 12 3 and 4mm posteriorly medially and proximally respectively averaged jcs parameters for the ankle were 43 degrees 9 degrees and 3 degrees for plantarflexion adduction and internal rotation respectively averaged maximal translation values for the ankle were 4 2 and 1mm anteriorly medially and proximally respectively

spiral self avoiding walks self avoiding walks with a spiral constraint on the square lattice are enumerated up to 40 steps numerical evidence suggests that they belong to a new universality class

an exact solution for a spiral self avoiding walk model on the triangular lattice exact results are derived for the number sn of spiral self avoiding walks with n steps on the triangular lattice in particle a closed form expression is presented for the generating function s x sigma n 1infinity snxn this result is used to establish a complete asymptotic expansion for sn which is valid as n to infinity

spiralling directed self avoiding walks a self avoiding walk model is proposed in which the walker performs two choice directed self avoiding walks while it may also make spiralling turns and thus visiting every quarter of the plane unlimited times the generating function for the square lattice is obtained and an asymptotic form for the number of n step walks an is derived the an of the model belongs to the same universality class with the two choice directed self avoiding walk however the model achieves a higher number of n step walks than the two choice directed self avoiding walk by a factor of c approximately 4 23609

validation of a three dimensional model of the knee three dimensional mathematical models of the tibio femoral joint require input of the geometry of articulating surfaces and ligament insertions and the mechanical properties of cartilage and ligaments this paper describes a validation of a knee model through a direct specimen related comparison between the knee model and the kinematics of four knee joint specimens from which the geometry data were used as input of the model the knee model is quasi static and is based on equilibrium of forces and moments the stiffness properties of the ligaments and articular cartilage were estimated on the basis of data reported in the literature the so called reference strains in the ligament bundles for the joint in extension were determined by using an optimization procedure minimizing the difference between the kinematics of the model and the kinematics of experimentally obtained flexion motions with an internally or an externally rotated tibia f 3 nm load a reasonable to good agreement between the model and the experimental kinematics could be obtained for internal external rotation laxity and the coupled translations and varu valgus rotation the disparity between model and experiment varied from knee to knee average deviations ranging from close to zero to 8 internal rotation deviation and from 5 mm posterior to 3 mm anterior position deviation the average anterior posterior laxities at both 20 and 90 flexion were within the variations reported in the literature although for each individual joint with some underestimation or overestimation it was concluded that the optimization procedure compensated for the lack of menisci and capsular structures by higher prestrains thereby overestimating the ligament forces despite the gross simplifica tions relative to the complex anatomy of the knee the present knee model can realistically simulate the passive motion characteristics of the human knee joint copyright 0 1996 elsevier science ltd

percolation under a spiralling constraint a new type of percolation process spiral percolation is considered in which each step of a percolation path proceeds either straight or in a specific rotational direction monte carlo simulation assuming a finite size scaling hypothesis has been performed on the square lattice to determine the spiral percolation threshold ps and the correlation length exponent nu s the values obtained are ps 0 730 or 0 004 and nu s 1 404 or 0 012

spiral bond animals ratio approach bond animals with a constraint of a given winding direction on the square lattice are enumerated up to 14 bonds numerical evidence further confirms the previous conjecture using the position space renormalisation group approach that they belong to a new universality class

flooding transition in the topography of toppling surfaces of stochastic and rotational sandpile models a continuous phase transition occurs in the topography of toppling surfaces of stochastic and rotational sandpile models when they are flooded with liquid say water the toppling surfaces are extracted from the sandpile avalanches that appear due to sudden burst of toppling activity in the steady state of these sandpile models though a wide distribution of critical flooding heights exists a critical point is defined by merging the flooding thresholds of all the toppling surfaces the criticality of the transition is characterized by power law distributionofislandareainthecriticalregime afinitesizescalingtheoryisdevelopedandverifiedbycalculating several new critical exponents the flooding transition is found to be an interesting phase transition and does not belong to the percolation universality class the universality class of this transition is found to depend on the degree of self affinity of the toppling surfaces characterized by the hurst exponent h and the fractal dimension df of critical spanning islands the toppling surfaces of different stochastic sandpile models are found to have a single hurst exponent whereas those of different rotational sandpile models have another hurst exponent as a consequence the universality class of different sandpile models remains preserved within the same symmetry of the models

functional knee axis based on isokinetic dynamometry data comparison of two methods mri validation and effect on knee joint kinematics this paper compares geometry based knee axes of rotation transepicondylar axis and geometric center axis and motion based functional knee axes of rotation faor two algorithms are evaluated to calculate faors gamage and lasenby s sphere fitting algorithm gl and ehrig et al s axis transformation algorithm sara calculations are based on 3d motion data acquired during isokinetic dynamometry aors are validated with the equivalent axis based on static mr images we quantified the difference in orientation between two knee axes of rotation as the angle between the projection of the axes in the transversal and frontal planes and the difference in location as the distance between the intersection points of the axes with the sagittal plane maximum differences between faors resulting from gl and sara were 5 7 and 15 4mm respectively maximum differences between faors resulting from gl or sara and the equivalent axis were 5 4 11 5mm and 8 6 12 8mm respectively differences between geometry based axes and ea are larger than differences between faor and ea both in orientation maximum 10 6 and location maximum 20 8mm knee joint angle trajectories and the corresponding accelerations for the different knee axes of rotation were estimated using kalman smoothing for the joint angles the maximum rms difference with the mri based equivalent axis which was used as a reference was 3 for the knee joint accelerations the maximum rms difference with the equivalent axis was 20 s 2 functional knee axes of rotation describe knee motion better than geometry based axes gl performs better than sara for calculations based on experimental dynamometry

number of spiral self avoiding loops on a triangular lattice exact results are derived for the number cn of spiral self avoiding loops with n steps on a triangular lattice a closed form expression for the generating function g x sigma n 1infinity cnxn is obtained for n to infinity cn increases as n6 31 104

a numerical solution of problems in calculus of variation using direct method and nonclassical parameterization in this paper a direct method for solving variational problems using nonclassical parameterization is presented a nonclassical parameterization based on nonclassical orthogonal polynomials is first introduced to reduce a variational problem to a nonlinear mathematical programming problem then using the lagrange multiplier technique the problem is converted to that of solving a system of algebraic equations illustrative examples are given to demonstrate the validity and applicability of the technique

on the symmetry classes of planar self avoiding walks we present new results on the class of anisotropic spiral walks in two dimensions we find that these are directed problems in the sense that the usual relation v 2vperpendicular to holds between the length scale exponents in contradistinction however they do not seem to fall in the usual directed universality class v 1 motivated by this the universality classes of self avoiding walks saw on the square lattice are discussed we argue that of the 84 models that exist by restricting the possible two step configurations there are four major categories with a total of seven generic types the importance of reflection symmetry in this classification is discussed