independent valued minimax pathological or beneficial minimax search which is used by most game playing programs is considered pathological when deeper searches produce worse evaluations than shallower ones this phenomenon was first observed in theoretical analyses under seemingly reasonable conditions it was most commonly explained by the lack of dependence between nearby positions in the analyses if nearby positions have similar values as is typically the case in real games the pathology no longer occurs in this paper we show that the pathology can be eliminated even without position value dependence by assigning enough different values to the positions and modeling the heuristic error as normally distributed noise that is independent of the depth in the game tree this leads to the conclusion that minimax is less prone to the pathology than was previously thought and indicates the importance of the number of different position values

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

efficient design of digital filters for 2 d and 3 d depth migration two and three dimensional 2 d and 3 d depth migration can be performed using 1 d and 2 d extrapolation digital filters respectively the depth extrapolation is done one frequency at a time by convolving the seismic wavefield with a complex valued frequency and velocity dependent digital filter this process requires the design of a complete set of extrapolation filters one filter for each possible frequency velocity pair instead of independently designing the frequency and velocity dependent filters an efficient procedure is introduced for designing a complete set of 1 d and 2 d extrapolation filters using transformations the problem of designing a desired set of migration filters is thus reduced to the design of a single 1 d filter which is then mapped to produce all the desired 1 d or 2 d migration filters the new design procedure has the additional advantage that both the 1 d and 2 d migration filters can be realized efficiently and need not have their coefficients precomputed or tabulated

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

mobius mellin and mathematical physics abstract we examine some results and techniques of analytic number theory which have application or potential application in mathematical physics we consider inversion formulae for lattice sums various transformations of infinite series and products functional equations and scaling relations with selected applications in electrostatics and statistical mechanics in the analysis the mellin transform and the riemann zeta function play a key role

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

number of two choice and spiral self avoiding loops the number of independent self avoiding walk saw loop configurations cn for fixed perimeter length n on the square lattice is fitted here to the scaling form cn approximately mu nnh for saws with various constraints for two choice saws extrapolation of enumeration results gives h 1 5 or 0 1 which falls in the same universality class as ordinary saws and exact calculations for spiral saws give h 3

the effect of mobility on minimaxing of game trees with random leaf values abstract random minimaxing introduced by beal and smith icca j 17 1994 3 9 is the process of using a random static evaluation function for scoring the leaf nodes of a full width game tree and then computing the best move using the standard minimax procedure the experiments carried out by beal and smith using random minimaxing in chess showed that the strength of play increases as the depth of the lookahead is increased we investigate random minimaxing from a combinatorial point of view in an attempt to gain a better understanding of the utility of the minimax procedure and a theoretical justification for the results of beal and smith s experiments the concept of domination is central to our theory intuitively one move by white dominates another move when choosing the former move would give less choice for black when it is black s turn to move and subsequently more choice for white when it is white s turn to move we view domination as a measure of mobility and show that when one move dominates another then its probability of being chosen is higher we then investigate when the probability of a good move relative to the probability of a bad move increases with the depth of search we show that there exist situations when increased depth of search is beneficial but that this is not always the case under the assumption that each move is either good or bad we are able to state sufficient conditions to ensure that increasing the depth of search increases the strength of play of random minimaxing if the semantics of the game under consideration match these assumptions then it is fair to say that random minimaxing appears to follow a reasonably intelligent strategy in practice domination does not always occur so it remains an open problem to find a more general measure of mobility in the absence of domination

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

partial order bounding a new approach to evaluation in game tree search abstract in computer game playing the established method for constructing an evaluation function uses a scalar value computed as a weighted sum of features this paper advocates the use of partial order evaluation and describes an efficient new search method called partial order bounding pob previous tree search algorithms using a partial order evaluation have attempted to propagate partially ordered values through the search tree which leads to many problems in practice such as the complexity of backing up sets of incomparable evaluations pob compares partially ordered values only in the leaves of a game tree and backs up boolean values through the tree a closely related new algorithm linear extension partial order bounding le pob uses a standard scalar alpha beta search with values from a suitably chosen linear extension of the partial order evaluation as an application the effectiveness of partial order evaluation is shown in the case of modeling capturing races called semeai in the game of go

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

3 d migration via mcclellan transformations on hexagonal grids two dimensional 2 d spatially varying convolutional filters for three dimensional 3 d depth extrapolation are efficiently designed and implemented by using mcclellan transformations although efficient mcclellan transformations only approximate circularly symmetric depth extrapolation filters the accuracy of the extrapolation filters can be increased only at an increase in the computational cost of implementing those filters by applying mcclellan transformations on a hexagonal sampling grid the accuracy of the extrapolation filters can be increased at a reduction in cost

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

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

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

designing stable extrapolators for explicit depth extrapolation of 2d and 3d wavefields using projections onto convex sets we have developed a robust algorithm for designing explicit depth extrapolation operators using the projections onto convex sets pocs method the operators are optimal in the sense that they satisfy all required extrapolation design characteristics in addition we propose a simple modification of the pocs algorithm modified pocs or mpocs that further enhances the stability of extrapolated wavefields and reduces the number of iterations required to design such operators to approximately 2 of that required for the basic pocs design algorithm various synthetic tests show that 25 coefficient 1d extrapolation operators which have 13 unique coefficients can accommodate dip angles up to 70 we migrated the seg eage salt model data with the operators and compare our results with images obtained via extrapolators based on modified taylor series and with wavefield extrapolation techniques such as phase shift plus interpolation pspi and split step fourier the mpocs algorithm provides practically stable

real time heuristic search abstract we apply the two player game assumptions of limited search horizon and commitment to moves in constant time to single agent heuristic search problems we present a variation of minimax lookahead search and an analog to alpha beta pruning that significantly improves the efficiency of the algorithm paradoxically the search horizon reachable with this algorithm increases with increasing branching factor in addition we present a new algorithm called real time a for interleaving planning and execution we prove that the algorithm makes locally optimal decisions and is guaranteed to find a solution we also present a learning version of this algorithm that improves its performance over successive problem solving trials by learning more accurate heuristic values and prove that the learned values converge to their exact values along every optimal path these algorithms effectively solve significantly larger problems than have previously been solvable using heuristic evaluation functions

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

an incremental negamax algorithm abstract in certain models of game trees with erroneous evaluation functions the minimax algorithm does not reduce errors even under favourable assumptions about the size of the errors and the frequency of their occurrence we present an incremental negamax algorithm which uses estimates of all nodes in the tree rather than only those of the leaves to determine the root value under the assumption of independently occurring and sufficiently small errors the algorithm is shown to have exponentially reduced error probabilities with respect to the height of the tree

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

closed form design and efficient implementation of variable digital filters with simultaneously tunable magnitude and fractional delay this paper proposes a closed form solution for designing variable one dimensional 1 d finite impulse response fir digital filters with simultaneously tunable magnitude and tunable fractional phase delay responses first each coefficient of a variable fir filter is expressed as a two dimensional 2 d polynomial of a pair of parameters called spectral parameters one is for independently tuning the cutoff frequency of the magnitude response and the other is for independently tuning fractional phase delay then the closed form error function between the desired and actual variable frequency responses is derived without discretizing any design parameters such as the frequency and the two spectral parameters finally the optimal solution for the 2 d polynomial coefficients can be easily determined through minimizing the closed form error function we also show that the resulting variable fir filter can be efficiently implemented by generalizing farrow structure to our two parameter case the generalized farrow structure requires only a small number of multiplications and additions for obtaining any new frequency characteristic which is particularly suitable for high speed tuning

the multi player version of minimax displays game tree pathology abstract it is widely believed that by searching deeper in the game tree the decision maker is more likely to make a better decision dana nau and others have discovered pathology theorems that show the opposite searching deeper in the game tree causes the quality of the ultimate decision to become worse not better the models for these theorems assume that the search procedure is minimax and the games are two player zero sum this report extends nau s pathology theorem to multi player game trees searched with maxn the multi player version of minimax thus two player zero sum game trees and multi player game trees are shown to have an important feature in common

critical behaviour of anisotropic spiral self avoiding walks the author considers three choice spiral and two choice spiral self avoiding walks saws on a square lattice for which the anisotropic constraints are such that for any nth step in the x and x direction the n 1 th step is forbidden to be in the clockwise or anticlockwise direction while for the nth step in the y and y direction there may or may not be any special constraints for the next step in the case of three choice spiral saws there is no such constraint other than self avoiding while in the case of two choice spiral saws with the nth step in the y and y directions the n 1 th step is forbidden to follow the same direction from exact enumeration results the author finds a new universality class having gamma 1 58 or 0 05 and nu 0 842 or 0 014 for such saws

implementation of 2d explicit depth extrapolation fir digital filters for 3d seismic volumes using singular value decomposition we propose a new scheme for implementing predesigned 2d complex valued wavefield extrapolation finite impulse response fir digital filters which are used for extrapolating 3d seismic wavefields the implementation is based on singular value decomposition svd of quadrantally symmetric 2d fir filters extrapolators to simplify the svd computations for such afilter impulse response structure we apply a special matrix transformation on the extrapolation fir filter impulse responses where we guarantee the retentionoftheirwavenumberphaseresponse unliketheexisting 2d fir filter implementation methods that are used for this geophysical application such as the mcclellan transformationoritsimprovedversion thisimplementationviasvdresults in perfect circularly symmetrical magnitude and phase wavenumber responses in this paper we also demonstrate that the svd method can save depending on the filter size more than 23 of the number of multiplications per output sample and approximately 62 of the number of additions per output sample when compared to direct implementation with quadrantal symmetry via true 2d convolution finally anapplicationtoextrapolationofaseismicimpulseisshown toproveourtheoreticalconclusions

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

best first minimax search we describe a very simple selective search algorithm for two player games called best first minimax it always expands next the node at the end of the expected line of play which determines the minimax value of the root it uses the same information as alpha beta minimax and takes roughly the same time per node generation we present an implementation of the algorithm that reduces its space complexity from exponential to linear in the search depth but at significant time cost our actual implementation saves the subtree generated for one move that is still relevant after the player and opponent move pruning subtrees below moves not chosen by either player we also show how to efficiently generate a class of incremental random game trees on uniform random game trees best first minimax outperforms alpha beta when both algorithms are given the same amount of computation on random trees with random branching factors best first outperforms alpha beta for shallow depths but eventually loses at greater depths we obtain similar results in the game of othello finally we present a hybrid best first extension algorithm that combines alpha beta with best first minimax and performs significantly better than alpha beta in both domains even at greater depths in othello it beats alpha beta in two out of three games

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

on anisotropic spiral self avoiding walks we report on a monte carlo study of so called two choice spiral self avoiding walks on the square lattice these have the property that their geometric size such as is measured by the radius of gyration scales anisotropically with exponent values that seem to defy rational fraction conjectures this polymer model was previously understood to be in a universality class different to ordinary self avoiding walks directed walks which are also anisotropic and symmetric spiral walks in two dimensions our monte carlo study concurs with those previous exact enumeration studies in that respect however we estimate substantially different values for the scaling exponents associated with the geometric size of the walks we give arguments that explain this difference in terms of a turning point in the local exponent values and in turn explain this by arguing for the existence of probable logarithmic corrections we also supply numerical evidence supporting a conjecture concerning the angle of anisotropy in the model

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

rooted spiral trees on hyper cubic lattices we study rooted spiral trees in 2 3 and 4 dimensions on a hyper cubic lattice using exact enumeration and monte carlo techniques on the square lattice we also obtain exact lower bound of 1 93565 on the growth constant series expansions give 1 3667 0 0010 and 0 6574 0 0010 on a square lattice with monte carlo simulations we get the estimates as 1 364 0 010 and 0 656 0 010 these results are numerical evidence against earlier proposed dimensional reduction by four in this problem in dimensions higher than two the spiral constraint can be implemented in two ways in either case our series expansion results do not support the proposed dimensional reduction

benefits of using multivalued functions for minimaxing abstract minimaxing has been very successful in game playing practice although a complete explanation of why it has been that successful has not yet been given in particular it has not been shown why it should be useful as it is in practice to use multivalued evaluation functions such functions have many distinct values as their result and can discriminate between positions according to the heuristic knowledge represented in these values in this paper we modify a basic pathological model by adding two assumptions regarding multivalued evaluation functions these assumptions non uniformity of error distribution and dependency of heuristic values directly relate to the properties of multivalued evaluation functions as used in practice simulation studies of our multivalued model have exhibited sharp error reductions for deeper searches using minimaxing this behavior corresponds to observations in practice the error reductions are primarily due to the improved evaluation quality as search depth increases this phenomenon of lower probability of static evaluation errors with increasing search depth is revealed through our model although the same evaluation function is used at all levels of the tree and although its general error probability is independent of the depth essentially with increasing search depth the evaluation function is more frequently used on such positions which can be more reliably evaluated by a multivalued function with the assumed properties this effect together with the ability to discriminate between positions of different goodness leads to the benefits of using multivalued evaluation functions of appropriate granularity for minimaxing

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

statistical mechanics of directed models of polymers in the square lattice directed square lattice models of polymers and vesicles have received considerable attention in the recent mathematical and physical sciences literature these are idealized geometric directed lattice models introduced to study phase behaviour in polymers and include dyck paths partially directed paths directed trees and directed vesicles models directed models are closely related to models studied in the combinatorics literature and are often exactly solvable they are also simplified versions of a number of statistical mechanics models including the self avoiding walk lattice animals and lattice vesicles the exchange of approaches and ideas between statistical mechanics and combinatorics have considerably advanced the description and understanding of directed lattice models and this will be explored in this review the combinatorial nature of directed lattice path models makes a study using generating function approaches most natural in contrast the statistical mechanics approach would introduce partition functions and free energies and then investigate these using the general framework of critical phenomena generating function and statistical mechanics approaches are closely related for example questions regarding the limiting free energy may be approached by considering the radius of convergence of a generating function and the scaling properties of thermodynamic quantities are related to the asymptotic properties of the generating function in this review the methods for obtaining generating functions and determining free energies in directed lattice path models of linear polymers is presented these methods include decomposition methods leading to functional recursions as well as the temperley method that is implemented by creating a combinatorial object one slice at a time a constant term formulation of the generating function will also be reviewed the thermodynamic features and critical behaviour in models of directed paths may be informative about the underlying properties that determine phase diagrams for wider classes of models including physical models of polymers of particular interest are adsorption and collapse transitions in models of polymers and copolymers the properties of thermodynamic quantities in those models are described by tricritical scaling this is reviewed for directed path models and the generating function approaches can be used to apply tricritical scaling to models of adsorbing inflating and collapsing directed lattice paths critical exponents for a variety of models can be obtained in this manner and with it a better understanding and a classification of the models

depth first iterative deepening an optimal admissible tree search abstract the complexities of various search algorithms are considered in terms of time space and cost of solution path it is known that breadth first search requires too much space and depth first search can use too much time and doesn t always find a cheapest path a depth first iterative deepening algorithm is shown to be asymptotically optimal along all three dimensions for exponential tree searches the algorithm has been used successfully in chess programs has been effectively combined with bi directional search and has been applied to best first heuristic search as well this heuristic depth first iterative deepening algorithm is the only known algorithm that is capable of finding optimal solutions to randomly generated instances of the fifteen puzzle within practical resource limits

on the nature of pathology in game searching abstract game playing programs usually process the estimates attached to game positions through repeated minimax operations as if these were true terminal payoffs this process introduces a spurious noise which degrades the quality of the decisions and in the extreme case may cause a pathological phenomenon the deeper we search the worse we play using a probabilistic game model this paper examines the nature of this distortion quantifies its magnitude determines the conditions when its damage is curtailed and explains why search depth pathology the extreme manifestation of minimax distortion is rarely observed in common games

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

3 d depth migration via mcclellan transformations three dimensional seismic wavefields may be extrapolated in depth one frequency at a time by two dimensional convolution with a circularly symmetric frequency and velocity dependent filter this depth extrapolation performed for each frequency independently lies at the heart of 3 d finite difference depth migration the computational efficiency of 3 d depth migration depends directly on the efficiency of this depth extrapolation mcclellan transformations provide an efficient method for both designing and implementing two dimensional digital filters that have a particular form of symmetry such as the circularly symmetric depth extrapolation filters used in 3 d depth migration given the coefficients of one dimensional frequency and velocity dependent filters used to accomplish 2 d depth migration mcclellan transformations lead to a simple and efficient algorithm for 3 d depth migration 3 d depth migration via mcclellan transformations is simple because the coefficients of two dimensional depth

linear complementarity problem a novel approach to design finite impulse response wavefield extrapolation filters abstractin this paper the problem of complex valued finite impulse response fir wavefield extrapolation filter design was considered as a linear complementarity problem lcp lcp is not an optimization technique because there is no objective function to optimize however quadratic programming one of the applications of lcp can be used to find an optimal solution for the 1d fir wavefield extrapolation filter quadratic programs are an extremely important source of applications of lcp in fact several algorithms for quadratic programs are based on lcp we found that fir wavefield extrapolation filter design problem can be written as a quadratic program and then finally to an equivalent lcp there are two families of algorithms available to solve for lcp 1 direct pivoting based algorithms and 2 indirect iterative algorithms in this study the lcp has been solved using direct and indirect algorithms to show the effectiveness of the proposed method the seg eage salt velocity model data ha

when is it better not to look ahead in situations where one needs to make a sequence of decisions it is often believed that looking ahead will help produce better decisions however it was shown 30 years ago that there are pathological situations in which looking ahead is counterproductive two long standing open questions are a what combinations of factors have the biggest influence on whether lookahead pathology occurs and b whether it occurs in real world decision making this paper includes simulation results for several synthetic game tree models and experimental results for three well known board games two chess endgames kalah with some modifications to facilitate experimentation and the 8 puzzle the simulations show the interplay between lookahead pathology and several factors that affect it and the experiments confirm the trends predicted by the simulation models the experiments also show that lookahead pathology is more common than has been thought all three games contain situations where it occurs

asymptotic properties of levelregular decision trees with randomly evaluated leaves game trees are an important model of decision making situations both in artificial intelligence and decision analysis the model most frequently investigated in theoretical research consists of a uniform tree of heighh and a constant branching factorb where the terminal positions are assigned the values of independent identically distributed random variables 1 3 10 our paper investigates two generalizations

the last player theorem game trees are an important model of decision making situations both in artificial intelligence and decision analysis but many of the properties of game trees are not well understood one of these properties is known as biasing when a minimax search is done to an odd search depth all moves tend to look good and when it is done to an even search depth all modes tend to look bad one explanation sometimes proposed for biasing is that whenever a player makes a move his position is strengthened and that the evaluation function used in the minimax search reflects this however the mathematical results in this paper suggest that biasing may instead be due to the errors made by the evaluation function

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

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

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

finite lattice calculation of the quantum mechanical n vector model the truncated quantum mechanical n vector model in one dimension is studied by means of the phenomenological renormalisation group prg and the block renormalisation group brg for general values of n particular emphasis is put on the extrapolation of the finite size results using the finite size scaling hypothesis accurate estimates of the critical properties can be made for 0 or approximately n or approximately 2 the two methods used are critically compared