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

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

on three dimensional self avoiding walk symmetry classes in two dimensions the universality classes of self avoiding walks saws on the square lattice restricted by allowing only certain two step configurations tscs to occur within each walk has been argued to be determined primarily by the symmetry of the set of allowed rules in three dimensions early work tentatively found one undirected universality class different to that of unrestricted saws on the simple cubic lattice this rule was a natural generalization of the square lattice spiral saw to three dimensions in this report we examine a variety of three dimensional saw models with different step restrictions carefully chosen so as to search for a connection between the symmetry of the rules and possible new universality classes a first analysis of the scaling of the radius of gyration suggests several universality classes including the one found earlier and perhaps some novel class es however a classification of these universality classes using the symmetries of the rules or other basic rule properties is not evident further analysis of the number of configurations and moment of inertia tensor suggests that in three dimensions the only non trivial or undirected universality class is that of unrestricted saws

exact solution of the partially directed compact lattice animal model the authors calculate exactly the number generating function for partially directed compact lattice animals on the square lattice for the fully directed model they report results of new numerical studies

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

an exact formula for the number of spiral self avoiding walks an exact closed form expression is obtained for the number sn of spiral self avoiding walks with n steps on the square lattice this result is used to derive a complete asymptotic expansion for sn which is valid as n to infinity

an exactly solvable self avoiding walks model a self avoiding walks model is studied in which the walker is in addition to the usual self avoiding condition restricted not to make any turn which will put the walker in a direction rotated more than phi max from any of the directions previously taken here phi max phi r where phi r 2 pi the smallest exterior angle of the unit cell of the lattice under consideration the generating function for the square lattice is obtained and various properties such as the number of n step walks the total number of steps to given directions and the mean square end to end distance are evaluated analytically the model exhibits characteristics reminiscent of one dimensional self avoiding walks in all directions while retaining the anisotropic effect of the direction of the first step

asymptotic properties of spiral self avoiding walks the authors consider the spiral self avoiding walk model recently introduced by privman 1983 on the basis of an analogy with the partitioning of the integers they argue that the number of n step spiral walks should increase asymptotically as rho square root n with rho a constant leading to an essential singularity in the generating function enumeration data to 65 terms indicates however that cn apparently varies as rho n alpha with alpha approximately 0 55 they also study the n dependence of the mean square end to end distance rn2 and of the mean rotation angle theta n for n step walks from series extrapolations they estimate that rn2 approximately n1 2 and theta n approximately n0 55

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

the asymptotic form for the number of spiral self avoiding walks the author investigates the asymptotic behaviour of the number sn of spiral self avoiding walks on the square lattice and show that sn approximately rho square root n with exp pi 312 or rho or exp 2 pi 312

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

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

the mean field theory in em procedures for blind markov random field image restoration a markov random field mrf model based em expectation maximization procedure for simultaneously estimating the degradation model and restoring the image is described the mrf is a coupled one which provides continuity inside regions of smooth gray tones and discontinuity at region boundaries constraints for the restoration problem which is in general ill posed the computational difficulty associated with the em procedure for mrfs is resolved by using the mean field theory from statistical mechanics an orthonormal blur decomposition is used to reduce the chances of undesirable locally optimal estimates experimental results on synthetic and real world images show that this approach provides good blur estimates and restored images the restored images are comparable to those obtained by a wiener filter in mean square error but are most visually pleasing

a mathematical analysis of the motion coherence theory in motion perception there are a number of important phenomena involving coherence examples include motion capture and motion cooperativity we propose a theoretical model called the motion coherence theory that gives a possible explanation for these effects 1 2 in this framework the aperture problem can also be thought of as a problem of coherence and given a similar explanation we propose the concept of a velocity field defined everywhere in the image even where there is no explicit motion information available theough a cost function the model imposes smoothness on the velocity field in a more general way than in previous theories in this paper we provide a detailed theoretical analysis of the motion coherence theory we discuss its relations with previous theories and show that some of them are approximations to it a second paper 3 provides extensions for temporal coherence and comparisons to psychophysical phenomena the theory applies to both short range and long range motion it places them in the same computational framework and provides a way to define interactions between the two processes

anisotropic spiral self avoiding walks manna has recently studied two extensions of spiral self avoiding walks on the square lattice with spiral constraints after east and west steps but not after north and south steps see ibid vol 17 p l899 1984 here the author calculates the connective constants for these two models

on the number of spiral self avoiding walks the authors consider the problem of spiral self avoiding walks as recently introduced by privman 1983 they prove that the number of n step spiral self avoiding walks is given by sn exp 2 pi n 3 12 n74 c 1 o 1 square root n where c pi 4 354 similar results for various subsets of these walks are also obtained

spiral self avoiding walks on the triangular lattice the authors study the behaviour of spiral self avoiding walks on the triangular lattice the spiral constraint simply says that no step in an anticlockwise direction may be taken imposing the additional constraint that steps may not deviate from the straight ahead direction by pi 3 defines model i previously solved by joyce and brak 1985 as well as by lin 1985 if deviations of 2pi 3 are forbidden this model is referred to as model ii while model iii allows deviations of both pi 3 and 2pi 3 one finds for both model ii and model iii that the number of n step self avoiding spirals is sn approximately c exp 2pi square root n log n 12 n13 4 where c phi2 768gamma5 phi model ii approximately 0 009 phi model iii approximately 0 16 and gamma 1 12 log 2 pi 2 the confluent logarithm is an additional feature not present in the simpler case of the square lattice and model i triangular spiral self avoiding walks use of two new results in the theory of partitions are made

spiralling self avoiding walks self avoiding walks which only step straight ahead or to the left are studied heuristic arguments generating function studies and extrapolation of few step enumerations are described exact asymptotic results are obtained it is found that these spiralling walks are in new universality class wherein the number of n step walks does not behave in a size extensive manner

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

comment on the number of spiral self avoiding walks the authors show that the number of spiral self avoiding walks on the square lattice is directly determined by the number of partitions of the integers the next term in the asymptotic expansion is found and numerous typographical errors in the original letter of guttmann and wormald 1984 are corrected

an exactly solvable self avoiding walks model ii triangular and honeycomb lattices a modified self avoiding walks model previously proposed for the square lattice is applied to the triangular and honeycomb lattices in the model the walker is restricted not to take any turn which will put the walker in a direction rotated by more than a certain angle phi max from any of the directions previously taken the generating functions are obtained and various quantities are evaluated exactly for all three square triangular and honeycomb lattices it is found that the model exhibits the characteristics reminiscent of one dimensional self avoiding walks in all directions while strongly retaining the anisotropic effect of the direction of the first step for the honeycomb lattice the authors have studied the model for phi max pi a model and phi max 3 2 pi b model unlike other walks the b model exhibits a special property the oscillations between even and odd steps in various quantities such as the number of n step walks do not decay as the number of steps increases