multiple comparison of several linear regression models research on multiple comparison during the past 50 years or so has focused mainly on the comparison of several population means several years ago spurrier considered the multiple comparison of several simple linear regression lines he constructed simultaneous confidence bands for all of the contrasts of the simple linear regression lines over the entire range when the models have the same design matrices this article extends spurrier s work in several directions first multiple linear regression models are considered and the design matrices are allowed to be different second the predictor variables are either unconstrained or constrained to finite intervals third the types of comparison allowed can be very flexible including pairwise many one and successive two simulation methods are proposed for the calculation of critical constants the methodologies are illustrated with examples

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

generalized differential riccati equation and indefinite stochastic lq control with cross term a stochastic indefinite linear quadratic lq optimal control problem with cross term in a finite time horizon is studied a new generalized differential riccati equation gdre is introduced it is shown that the solvability of the gdre is equivalent to the solvability of the indefinite stochastic lq problem furthermore all of the optimal controls including feedback and open loop can be identified via the solution to the gdre an example is presented to illustrate the theory obtained

stochastic linear quadratic control with conic control constraints on an infinite time horizon this paper is concerned with a stochastic linear quadratic lq control problem in the infinite time horizon where the control is constrained in a given arbitrary closed cone the cost weighting matrices are allowed to be indefinite and the state is scalar valued first the mean square conic stabilizability of the system is defined which is then characterized by a set of simple conditions involving linear matrix inequalities lmis next the issue of well posedness of the underlying optimal lq control which is necessitated by the indefiniteness of the problem is addressed in great detail and necessary and sufficient conditions of the well posedness are presented on the other hand to address the lq optimality two new algebraic equations a la riccati called extended algebraic riccati equations eares along with the notion of their stabilizing solutions are introduced for the first time optimal feedback control as well as the optimal value are explicitly derived in terms of the stabilizing solutions to the eares moreover several cases when the stabilizing solutions do exist are discussed and algorithms of computing the solutions are presented finally numerical examples are provided to illustrate the theoretical results established

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

constrained stochastic lqc a tractable approach despite the celebrated success of dynamic programming for optimizing quadratic cost functions over linear systems such an approach is limited by its inability to tractably deal with even simple constraints in this paper we present an alternative approach based on results from robust optimization to solve the stochastic linear quadratic control slqc problem in the unconstrained case the problem may be formulated as a semidefinite optimization problem sdp we show that we can reduce this sdp to optimization of a convex function over a scalar variable followed by matrix multiplication in the current state thus yielding an approach that is amenable to closed loop control and analogous to the riccati equation in our framework we also consider a tight second order cone socp approximation to the sdp that can be solved much more efficiently when the problem has additional constraints both the sdp and socp are tractable in the presence of control and state space constraints moreover compared to the riccati approach they provide much greater control over the stochastic behavior of the cost function when the noise in the system is distributed normally

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

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

dynamic mean variance portfolio selection with no shorting constraints this paper is concerned with mean variance portfolio selection problems in continuous time under the constraint that short selling of stocks is prohibited the problem is formulated as a stochastic optimal linear quadratic lq control problem however this lq problem is not a conventional one in that the control portfolio is constrained to take nonnegative values due to the no shorting restriction and thereby the usual riccati equation approach involving a completion of squares does not apply directly in addition the corresponding hamilton jacobi bellman hjb equation inherently has no smooth solution to tackle these difficulties a continuous function is constructed via two riccati equations and then it is shown that this function is a viscosity solution to the hjb equation solving these riccati equations enables one to explicitly obtain the efficient frontier and efficient investment strategies for the original mean variance problem an example illustrating these results is also presented

equivalence of regression curves this article investigates the problem whether the difference between two parametric models m1 m2 describing the relation between a response variable and several covariates in two different groups

the number of convex polygons on the square and honeycomb lattices a subset of the set of self avoiding polygons sap embeddable on the square lattice which display the property of convexity is defined an algorithm for their enumeration is developed and from the available series coefficients the exact generating function is found the singularity structure appears similar to that of the unsolved sap problem but with different critical exponents and critical points the enumeration of convex polygons allows the extension of the existing series for the square lattice sap by one term for the honeycomb lattice similar results have been obtained despite a less natural definition of convexity

confidence intervals associated with tests for bioequivalence summary previously the connection between bioequivalence tests and their associated confidence intervals was not well understood in fact the oc level two one sided tests approach was thought to be associated with a 1 20c confidence interval in this paper we build up connections which allow us to construct 1 oc confidence intervals from cx level tests and vice versa when applied to the cc level two one sided tests the resultant 1 oc confidence intervals are properly contained in westlake s 1 oc symmetric interval 1976 our approach is readily generalized to different settings including the nonparametric setting the ratio parameter setting and the repeated confidence interval setting

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

stochastic receding horizon control of constrained linear systems with state and control multiplicative noise we develop a receding horizon control approach to stochastic linear systems with control and state multiplicative noise that also contain constraints our receding horizon formulation is based upon an on line optimization that utilizes open loop plus linear feedback and is solved as a semi definite programming problem we also provide a characterization of stability performance and constraint satisfaction properties of the receding horizon controlled system under a specific choice of terminal weight and terminal constraint a simple numerical example is used to illustrate the approach

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

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

simultaneous confidence bands for a percentile hyper plane with covariates constrained in a restricted range abstract a simultaneous confidence band provides useful information about the plausible range of the unknown regression model in several recent papers confidence bands have been used for various inferential purposes see for example liu et al 2004 2005 2009 2010 2012 2014 liu and hayter 2007 1 liu and lin 2009 piegorsch et al 2005 han et al 2015 and zhou 2017 the construction of simultaneous confidence bands has a long history going back to working and hotelling 1929 2 and is often difficult when the region over which a confidence band is required is restricted and the number of predictor variables is greater than one in this paper we have developed methods to construct simultaneous confidence bands for a percentile hyper plane these methods allow us to construct the bands over arbitrary intervals either finite or infinite they also allow us to construct the type i and type ii bands in order to get an idea of the efficiency of these bands we have also developed the confidence sets for the bands and have compared the bands under the minimum volume confidence set optimality criterion the proposed methods are illustrated by an example

assessing nonsuperiority noninferiority or equivalence when comparing two regression models over a restricted covariate region in many scientific problems the purpose of the comparison of two regression models which describe the relationship between a same response variable and several same covariates for two different groups is to demonstrate that one model is no higher than the other by a negligible amount or to demonstrate that the models have only negligible differences and so they can be regarded as describing practically the same relationship between the response variable and the covariates in this article methods based on one sided pointwise confidence bands are proposed for assessing the nonsuperiority of one model to the other and for assessing the equivalence of two regression models examples from qt qtc study and from drug stability study are used to illustrate the methods

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

pooling batches in drug stability study by using constant width simultaneous confidence bands one important study objective in drug stability studies is to estimate the shelf life of a drug a key statistical problem involved in this is how to assess the practical equivalence of different batches of the same drug so that different batches can be subgrouped to produce a single shelf life for the drug in this paper constant width simultaneous confidence bands are proposed to quantify the magnitude of difference between different batches with a particular view to establish the practical equivalence of different batches this approach is suitable for the situation that the intercepts and slopes of the regression lines for the batches cannot be assumed to be equal it is shown how constant width simultaneous confidence bands can be easily constructed for the multiple comparison of several general linear regression models in particular it is shown that constant width simultaneous confidence bands have a better chance to establish the equivalence than and so are preferable to the hyperbola shaped simultaneous confidence bands considered j am stat assoc 2004 99 395 403

pooling data for stability studies testing the equality of batch degradation slopes pharmaceutical products are routinely monitored for their stability over time stability studies generally consist of a random sample of dosage units e g tablets capsules vials from a batch or several batches placed in a storage room and periodically assayed for their drug content the degradation of the drug product is modeled and according to the guideline for submitting documentation for stability studies of human drugs and biologics food and drug administration 1987 the shelf life is calculated as the time point at which the lower 95 confidence limit about the fitted regression line crosses the lowest acceptable limit for drug content frequently 90 of the labeled amount when multiple batches are manufactured preliminary testing for any batch differences both slope and intercept should precede pooling stability data from all batches in the analysis the guideline recommends a level of significance of 25 for such preliminary testing based on the work described by bancroft 1964 biometrics 20 427 442 using such a large significance level helps ensure that the power of the test for the batch differences is sufficiently high this paper presents an approach whereby the power of the test is fixed and the significance level of the test needed to obtain this power is calculated from the data if the observed significance level does not achieve the calculated significance level then the data can be pooled examples will illustrate the relative performance of the fda guideline and the proposed procedure

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

stochastic control for linear systems driven by fractional noises this paper is concerned with optimal control of stochastic linear systems involving fractional brownian motion fbm first as a prerequisite for studying the underlying control problems some new results on stochastic integrals and stochastic differential equations associated with fbm are established then three control models are formulated and studied in the first two models the state is scalar valued and the control is taken as markovian either the problems are completely solved based on a riccati equation for model 1 where the cost is a quadratic functional on state and control variables or optimality is characterized for model 2 where the cost is a power functional the last control model under investigation is a general one where the system involves the stratonovich integral with respect to fbm the state is multidimensional and the admissible controls are not limited to being markovian a new riccati type equation which is a backward stochastic differential equation involving both fbm and normal brownian motion is introduced optimal control and optimal value of the model are explicitly obtained based on the solution to this riccati type equation

stochastic linear quadratic control via semidefinite programming we study stochastic linear quadratic lq optimal control problems over an infinite time horizon allowing the cost matrices to be indefinite we develop a systematic approach based on semidefinite programming sdp a central issue is the stability of the feedback control and we show this can be effectively examined through the complementary duality of the sdp furthermore we establish several implication relations among the sdp complementary duality the generalized riccati equation and the optimality of the lq control problem based on these relations we propose a numerical procedure that provides a thorough treatment of the lq control problem via primal dual sdp it identifies a stabilizing feedback control that is optimal or determines that the problem possesses no optimal solution for the latter case we develop an epsilon approximation scheme that is asymptotically optimal

relationship between backward stochastic differential equations and stochastic controls a linear quadratic approach it is well known that backward stochastic differential equations bsdes stem from the study on the pontryagin type maximum principle for optimal stochastic controls a solution of a bsde hits a given terminal value which is a random variable by virtue of an it additional martingale term and an indefinite initial state this paper attempts to explore the relationship between bsdes and stochastic controls by interpreting bsdes as some stochastic optimal control problems more specifically associated with a bsde a new stochastic control problem is introduced with the same dynamics but a definite given initial state the martingale term in the original bsde is regarded as the control and the objective is to minimize the second moment of the difference between the terminal state and the terminal value given in the bsde this problem is solved in a closed form by the stochastic linear quadratic lq theory developed recently the general result is then applied to the black scholes model where an optimal mean variance hedging portfolio is obtained explicitly in terms of the option price finally a modified model is investigated where the difference between the state and the expectation of the given terminal value at any time is taken into account

indefinite stochastic linear quadratic control and generalized differential riccati equation a stochastic linear quadratic lq control problem is indefinite when the cost weighting matrices for the state and the control are allowed to be indefinite indefinite stochastic lq theory has been extensively developed and has found interesting applications in finance however there remains an outstanding open problem which is to identify an appropriate riccati type equation whose solvability is it equivalent to the solvability of the indefinite stochastic lq problem this paper solves this open problem for lq control in a finite time horizon a new type of differential riccati equation called the generalized differential riccati equation is introduced which involves algebraic equality inequality constraints and a matrix pseudoinverse it is then shown that the solvability of the generalized riccati equation is not only sufficient but also necessary for the well posedness of the indefinite lq problem and the existence of optimal feedback open loop controls moreover all of the optimal controls can be identified via the solution to the riccati equation an example is presented to illustrate the theory developed

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

the efficiency of simulation based multiple comparisons a frequently encountered problem in practice is that of simultaneous interval estimation of p linear combinations of a parameter beta in the setting of or equivalent to a univariate linear model this problem has been solved adequately only in a few settings when the covariance matrix of the estimator is diagonal in other cases conservative solutions can be obtained by the methods of scheffe bonferroni or sidak 1967 journal of the american statistical association 62 626 633 here we investigate the efficiency of using a simulated critical point for exact intervals which has been suggested before but never put to serious test we find the simulation based method to be completely reliable and essentially exact sample size savings are substantial in our settings 3 19 over the sidak method 4 37 over the bonferroni method and 27 33 over the scheffe method we illustrate the efficiency and flexibility of the simulation based method with case studies in physiology and marine ecology

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

stochastic linear quadratic regulators with indefinite control weight costs ii in part i of this paper s chen x li and x zhou siam j control optim 36 1998 pp 1685 1702 an optimization model of stochastic linear quadratic regulators lqrs with indefinite control cost weighting matrices is proposed and studied in this sequel the problem of solving lqr models with system diffusions dependent on both state and control variables which is left open in part i is tackled first the solvability of the associated stochastic riccati equations sres is studied in the normal case namely all the state and control weighting matrices and the terminal matrix in the cost functional are nonnegative definite with at least one positive definite which in turn leads to an optimal state feedback control of the lqr problem in the general indefinite case the problem is decomposed into two optimal lqr problems one with a forward dynamics and the other with a backward dynamics the well posedness and solvability of the original lqr problem are then obtained by solving these two subproblems and an optimal control is explicitly constructed examples are presented to illustrate the results

indefinite stochastic riccati equations this paper is concerned with stochastic riccati equations sres which are a class of matrix valued nonlinear backward stochastic differential equations bsdes the sres under consideration are in general indefinite in the sense that certain parameter matrices are indefinite this kind of equations arises from the stochastic linear quadratic lq optimal control problem with random coefficients and indefinite state and control weighting costs the latter having profound implications in both theory and applications while the solvability of the sres is the key to solving the indefinite stochastic lq control it remains in general an extremely difficult open problem this paper attempts to solve the problem of existence and uniqueness of solutions to the indefinite sres for a number of special yet important cases

estimating drug shelf life with random batches the problem of assessing the shelf lives of drug products in the market is considered we focus on the situation where the drug characteristic of interest has a linear relationship over time and batch to batch variation is present two methods for assessing shelf lives are proposed based on the weighted least squares method under a regression model with random coefficients an application to some stability data from the pharmaceutical industry is presented

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

exact confidence bounds for all contrasts of three or more regression lines abstract it is desired to compare 3 treatments under the assumption of iid normal errors it is well known that the scheffe method produces exact simultaneous confidence bounds for all contrasts of the treatment means furthermore it is known that the scheffe method is conservative when one desires confidence bounds for a specific subset of contrasts of means exact methods such as those due to tukey and dunnett yield tighter bounds than the scheffe method for specific subsets of contrasts of means in this article multiple comparisons of the treatments are done not in terms of their means but rather in terms of a parametric function the parametric function of interest is the simple linear regression model e y x it is desired to find simultaneous confidence bounds for all contrasts of the simple linear regression models although the scheffe method can be used to find such bounds this is extremely conservative the union intersection method is used to develop simultaneous confidence bou

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