an cal o n level set method for eikonal equations a propagating interface can develop corners and discontinuities as it advances level set algorithms have been extensively applied for the problems in which the solution has advancing fronts one of the most popular level set algorithms is the so called fast marching method fmm which requires total cal o n log 2n operations where n is the number of grid points the article is concerned with the development of an cal o n level set algorithm called the group marching method gmm the new method is based on the narrow band approach as in the fmm however it is incorporating a correction by iteration strategy to advance a group of grid points at a time rather than sorting the solution in the narrow band to march forward a single grid point after selecting a group of grid points appropriately the gmm advances the group in two iterations for the cost of slightly larger than one iteration numerical results are presented to show the efficiency of the method applied to the eikonal equation in two and three dimensions

monte carlo study of the point for collapsing trees we investigate the collapse transition of lattice trees with nearest neighbor attraction in two and three dimensions two methods are used 1 a stochastic optimization process of the robbins monro type which is designed solely to locate the maximum value of the specific heat and 2 umbrella sampling which is designed to sample data over a wide temperature range as well as to combat the quasiergodicity of metropolis algorithms in the collapsed phase we find good evidence that the transition is second order with a divergent specific heat and that the divergence of the specific heat coincides with the metric collapse

on the finite size scaling of clusters in compact directed percolation the exact finite size scaling properties of clusters in compact directed percolation on a square lattice are derived the results are implicit in previous work on the enumeration of staircase polygons but their explicit form has not been presented as such before the analysis provides important insights into the nature of this type of percolation transition

rigorous results for the number of convex polygons on the square and honeycomb lattices the generating functions for the number of convex polygons on the square and honeycomb lattices are derived rigorously these functions were found by guttmann and enting 1988 their calculation is based on the series expansions up to the 64th order and is nonrigorous the asymptotic form of the mean squared radius of gyration of n step convex polygons on the square lattice is determined and the critical exponent v is 1

exact scaling functions for self avoiding loops and branched polymers it is shown that a recently conjectured form for the critical scaling function for planar self avoiding polygons weighted by their perimeter and area also follows from an exact renormalization group flow into the branched polymer problem combined with the dimensional reduction arguments of parisi and sourlas the result is generalized to higher order multicritical points yielding exact values for all their critical exponents and exact forms for the associated scaling functions

a new two variable generating function for convex self avoiding polygons the width of a self avoiding polygon on the square lattice is defined as the minimal horizontal or vertical distance between two of its parallel edges if the polygons are convex this distance is internal the perimeter generating functions for such convex polygons whose widths exceed a threshold can be given explicitly from these expressions a two variable width and perimeter generating function can be constructed the corresponding phase diagram shows two types of critical behaviour which meet at a tricritical point

convex polyominoes and heaps of segments builds a one to one correspondence between some polyominoes called parallelogram polyominoes and some heaps of segments then the author derives explicit expressions for the generating function of convex polyominoes according to their height width and area the author also enumerates two subsets of convex polyominoes namely the directed and convex polyominoes and the parallelogram polyominoes according to these three parameters generating functions of polyominoes having a fixed width are also studied

an eulerian method for capturing caustics a robust numerical method for the localization of caustics is proposed for general hamiltonians it is based on the direct resolution of a system of partial differential equations obtained through a local change of the time variable in the hamilton jacobi equation and complemented by a set of transport equations numerical results 1 to 3 d are presented

geometric optics in a phase space based level set and eulerian framework we introduce a level set approach for ray tracing and the construction of wavefronts in geometric optics this is important in a wide variety of applications in wave propagation our approach automatically handles the multivalued solutions that appear and automatically resolves the wavefronts this is achieved through solving for the bicharacteristic strips whose projection to spatial space gives the wavefronts in a reduced phase space under a eulerian and partial differential equation based framework the bicharacteristic strips are represented using a level set approach for handling higher codimensional objects and the partial differential equations responsible for the evolution are reduced forms of the liouville equations results for the two dimensional case for constant and variable indices of refraction are shown and compared to those of other current methods in the field results are also introduced to show the ability to handle reflection and to extend the method to the three dimensional case

exact scaling behavior of partially convex vesicles we solve analytically for the perimeter area generating functions for two models of vesicles while from the solution of the first model staircase polygons one can easily extract the asymptotic scaling behavior the exact solution of the second column convex polygons is difficult to analyze this leads us to apply a recently developed method for deriving the scaling behavior indirectly utilizing a set of nonlinear differential equations one result of this work is a nontrivial confirmation of the scaling universality hypothesis

difference equations in statistical mechanics i cluster statistics models a review and some new results are presented for several cluster statistics models solutions of which can be reduced to difference equations mathematical techniques suitable for solving these equations are surveyed

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

computation of the frequency response gains and h norm of a sampled data system abstract this paper presents a method for computing the frequency response gains and h norm l 2 induced norm of a sampled data system composed of continuous time and discrete time systems together with a sampler and a hold the method is based on the notion of fr operator and uses only such elementary frequency domain notions as conventional z transformation pulse transfer function and impulse modulation formula while the numerical computations can be carried out using state space equations this method turns out to have a natural parallelism to the usual methods for continuous time and pure discrete time systems and the frequency response gain at each frequency can be computed to any degree of accuracy in a numerically reliable fashion

interacting columns generating functions and scaling exponents a model of self interacting columns related to models of partially directed walks and to histogram polygons is considered the generating function of this model is found in terms of q deformed bessel functions using a functional recursion scheme a transition related to a deflation inflation transition seen in staircase polygons or to a rough smooth transition in a solid on solid model is found and its scaling exponents are found in the context of a tricritical scaling analysis if the columns are also interacting with the horizontal axis then the inflated or smooth phase is also found to undergo an adsorption transition a special point exists where the model is critical with respect to both an adsorption transition and a deflation inflation transition

area weighted moments of convex polygons on the square lattice when constructing series expansions for the zero field partition function of the q state potts model on the square lattice using the finite lattice method the series can be extended by one term by adding a correction of q 1 times the number of convex polygons of an appropriate order by virtue of duality such corrections apply for both high and low temperature series for low temperature series corrections to the field dependent partition function are of the same order in the temperature variable as the zero field correction the general corrections are given by area weighted moments of the number of convex polygons the authors obtain conjectured closed form expressions for the two generating functions that give corrections to the first two field derivatives of the low temperature partition function

scaling behaviour of two dimensional polygon models exactly solvable two dimensional polygon models counted by perimeter and area are described by q algebraic functional equations we provide techniques to extract the scaling behaviour of these models up to arbitrary order and apply them to some examples these are then used to analyze the unsolved model of self avoiding polygons where we numerically confirm predictions about its scaling function and its first two corrections to scaling

exact solution of the convex polygon perimeter and area generating function an explicit expression is derived for the three variable generating function p x y z sigma m or 1 sigma n or 1 sigma r or 1 x2ny2mzrcn m r where cn m r is the number of convex polygons with horizontal width n vertical height m and area r

area distribution for directed random walks we study the probability distribution for the area under a directed random walk in the plane the walk can serve as a simple model for avalanches based on the idea that the front of an avalanche can be described by a random walk and the size is given by the area enclosed this model captures some of the qualitative features of earthquakes avalanches and other self organized critical phenomena in one dimension by finding nonlinear functional relations for the generating functions we calculate directly the exponent in the size distribution law and find it to be 4 3

self avoiding polygons on the square lattice we have developed an improved algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 90 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 529 27 1 biased and the critical exponent 0 500 0005 10 unbiased the critical point is indistinguishable from a root of the polynomial 581x4 7x2 13 0 an asymptotic expansion for the coefficients is given for all n there is strong evidence for the absence of any non analytic correction to scaling exponent

a scaling theory of the collapse transition in geometric cluster models of polymers and vesicles much effort has been expended in the past decade to calculate numerically the exponents at the collapse transition point in walk polygon and animal models the crossover exponent phi has been of special interest and sometimes is assumed to obey the relation 2 alpha 1 phi with the alpha the canonical thermodynamic exponent that characterizes the divergence of the specific heat the reasons for the validity of this relation are not widely known the authors present a scaling theory of collapse transitions in such models the free energy and canonical partition functions have finite length scaling forms whilst the grand partition function has a tricritical scaling form the link between the grand and canonical ensembles leads to the above scaling relation they then comment on the validity of current estimates of the crossover exponent for interacting self avoiding walks in two dimensions and propose a test involving the scaling relation which may be used to check these values

generating functions for classes of self avoiding polygons on the square lattice abstract a transfer matrix technique is presented which allows the explicit calculation of the generating functions for those self avoiding polygons on the square lattice which have a perimeter equal to or two larger than the perimeter of their bounding rectangles

convex polyominoes and algebraic languages uses the so called dsv methodology that links some enumeration problems to the theory of algebraic languages to get a system of q difference equations involving the generating functions of convex polyominoes of convex and directed polyominoes and of parallelogram polyominoes according to their height width and area then the author shows various applications of this system

voronoi diagram and medial axis algorithm for planar domains with curved boundaries i theoretical foundations in this first installment of a two part paper the underlying theory for an algorithm that computes the voronoi diagram and medial axis of a planar domain bounded by free form polynomial or rational curve segments is presented an incremental approach to computing the voronoi diagram is used wherein a single boundary segment is added to an existing boundary segment set at each step the introduction of each new segment entails modifying the voronoi regions of the existing boundary segments and constructing the voronoi region of the new segment we accomplish this by i computing the bisector of the new segment with each of the current boundary segments ii updating the voronoi regions of the current boundary segments by partitioning them with these bisectors and iii constructing the voronoi region of the new segment as a union of regions obtained from the partitioning in ii when all boundary segments are included and their voronoi regions have been constructed the voronoi diagram of the boundary is obtained as the union of the voronoi polygons for each boundary segment to construct the medial axis of a planar domain we first compute the voronoi diagram of its boundary the medial axis is then obtained from the voronoi diagram by i removing certain edges of the voronoi diagram that do not belong to the medial axis and ii adding certain edges that do belong to the medial axis but are absent from the voronoi diagram unambiguous characterizations for edges in both these categories are given details of algorithms based on this theory are deferred to the second installment of this two part paper

jacobi s last geometric statement extends to a wider class of liouville surfaces numerical evidence is presented which strongly suggests that jacobi s last geometric statement that the conjugate locus from a point has exactly four cusps and the corresponding cut locus consists of only one topological segment holds for compact real analytic liouville surfaces diffeomorphic to s 2 if the gaussian curvature is everywhere positive and has exactly six critical points these being two saddles two global minima and two global maxima as is the case for an ellipsoid our experiments suggest that this is a sufficient rather than a necessary condition furthermore for compact real analytic liouville surfaces diffeomorphic to s 2 upon which the gaussian curvature can be negative but has exactly six critical points these being two saddles two global minima and two global maxima it appears that the cut locus is always a subarc of a line given by x 1 const or x 2 const where x 1 x 2 are canonical coordinates with respect to which the metric has the form f 1 x 1 f 2 x 2 dx 2 1 dx 2 2 in the case of an ellipsoid these curves are lines of curvature

on an exactly solvable compact cluster growth model on a square lattice we consider a cluster growth model on a square lattice that is in the compact directed percolation universality class the model is exactly solvable as regards its static cluster properties and expressions are given for various quantities of interest such as the mean perimeter length and the mean cluster size the results also provide new information on area perimeter generating functions for a class of self avoiding polygons on square lattices

area width scaling in generalised motzkin paths abstract we consider a generalised version of motzkin paths where horizontal steps have length l with l being a fixed positive integer we first give the general functional equation for the area width generating function of this model using a heuristic ansatz we then derive the area width scaling behaviour in terms of a scaling function in one variable for the special cases of dyck standard motzkin and schroder paths before generalising our approach to arbitrary l we then rigorously derive the tricritical scaling of schroder paths by applying the generalised method of steepest descents to the known exact solution for their area width generating function our results show that for dyck and schroder paths the heuristic scaling ansatz reproduces the rigorous results

the tricritical behavior of self interacting partially directed walks we present the thermodynamics of two variations of the interacting partially directed self avoiding walk problem by discussing versions where the length of the walks assume real as well as a integral values while the discrete model has been considered previously to varying degrees of success the continuous model we now define has not the examination of the continuous model leads to theexact derivation of several exponents for the discrete model some of these exponents can be calculated using a continued fraction representation for both models the crossover exponent is found to be 2 3 moreover we confirm the tricritical nature of the collapse transition in the generalized ensemble and calculate the full scaling form of the generating function additionally the similarities noticed previously to other models but left unexplored are explained with the aid of necklacing arguments

scaling function and universal amplitude combinations for self avoiding polygons we analyse new data for self avoiding polygons saps on the square and triangular lattices enumerated by both perimeter and area providing evidence that the scaling function is the logarithm of an airy function the results imply universal amplitude combinations for all area moments and suggest that rooted saps may satisfy a q algebraic functional equation

efficient algorithms for globally optimal trajectories we present serial and parallel algorithms for solving a system of equations that arises from the discretization of the hamilton jacobi equation associated to a trajectory optimization problem of the following type a vehicle starts at a prespecified point x sub o and follows a unit speed trajectory x t inside a region in spl rscr sup m until an unspecified time t that the region is exited a trajectory minimizing a cost function of the form spl int sub 0 sup t r x t dt q x t is sought the discretized hamilton jacobi equation corresponding to this problem is usually solved using iterative methods nevertheless assuming that the function r is positive we are able to exploit the problem structure and develop one pass algorithms for the discretized problem the first algorithm resembles dijkstra s shortest path algorithm and runs in time o n log n where n is the number of grid points the second algorithm uses a somewhat different discretization and borrows some ideas from a variation of dial s shortest path algorithm 1969 that we develop here it runs in time o n which is the best possible under some fairly mild assumptions finally we show that the latter algorithm can be efficiently parallelized for two dimensional problems and with p processors its running time becomes o n p provided that p o spl radic n log n

asymptotic behavior of inflated lattice polygons we study the inflated phase of two dimensional lattice polygons with fixed perimeter n and variable area associating a weight exp pa jb to a polygon with area a and b bends for convex and column convex polygons we calculate the average area for positive values of the pressure for large pressures the area has the asymptotic behaviour langle a rangle a max 1 k j tilde p 2 mathcal o rho tilde p where tilde p pn gg 1 and 1 the constant k j is found to be the same for both types of polygons we argue that self avoiding polygons should exhibit the same asymptotic behavior for self avoiding polygons our predictions are in good agreement with exact enumeration data for j 0 and monte carlo simulations for j 0 we also study polygons where self intersections are allowed verifying numerically that the asymptotic behavior described above continues to hold

exact solution of the row convex polygon perimeter generating function an explicit expression is obtained for the perimeter generating function g y sigma n or 2 any2n for row convex polygons on the square lattice where an is the number of 2n step row convex polygons an asymptotic expression for an approximately a mu nn 3 2 is obtained where mu 3 22 and a are given the authors also show that the generating function is an algebraic function and that it satisfies an inhomogeneous linear differential equation of degree three

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

on directed compact percolation near a damp wall a percolation probability for directed compact percolation near a damp wall which interpolates between the previously examined cases is derived exactly we find that the critical exponent 2 in common with the dry wall rather than the value previously found in the wet wall and bulk cases the solution is found via a mapping to a particular model of directed walks we evaluate the exact generating function for this walk model which is also related to the asep model of traffic flow we compare the underlying mathematical structure of the various cases previously considered and this one by reviewing the common framework of solution via the mapping to different directed walk models

compact directed percolation with modified boundary rules a forest fire model a modified version of compact directed percolation on a square lattice is examined in the context of a model problem for the spread of forest fires the modification relates to conditioning the extent of the fire spread along the forest boundaries exact expressions are given for the mean perimeter length and the mean size of the damaged forest under such conditioning

the theory of caustics and wave front singularities with physical applications this is intended as an introduction to and review of the theory of lagrangian and legendrian submanifolds and their associated maps developed by arnold and his collaborators the theory is illustrated by applications to hamilton jacobi theory and the eikonal equation with an emphasis on null surfaces and wave fronts and their associated caustics and singularities

roof edge detection using regularized cubic b spline fitting abstract a scheme employing one dimensional 1 d regularized cubic b spline rcbs fitting g chen and y h yang ieee trans systems man cybernet 25 636 693 task of step edge detection the regularized fitting is transformed into a quadratic energy equation to simplify the computation this scheme however has three major limitations it is non linear has a limited accuracy and is computationally expensive this paper presents a modified scheme which overcomes these limitations the modified scheme employs the 1 d rcbs fitting on the horizontal and vertical orientations of a window of an image to generate two 1 d signals which provide sufficient information about the local property of the sub image for roof edge detection experimental results show that the scheme of roof edge detection is very sensitive to small signals

algebraic languages and polyominoes enumeration abstract in this paper the use of algebraic languages theory in solving an open problem in combinatorics is shown by constructing a bijection between convex polyominoes and words of an algebraic language and by solving the corresponding algebraic system we prove that the number of convex polyominoes with perimeter 2n 8 is 2n 11 4 n 4 2n 1 2n n

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

on the asymptotics of the finite perimeter partition function of two dimensional lattice vesicles we derive the dominant asymptotic form and the order of the correction terms of the finite perimeter partition function of self avoiding polygons on the square lattice which are weighted according to their area a as qa in the inflated regime q 1 the approach q 1 of the asymptotic form is examined

anisotropic step mutual contact and area weighted festoons and parallelogram polyominoes on the triangular lattice we present results for the generating functions of polygons and more general objects that can touch constructed from two fully directed walks on the infinite triangular lattice enumerated according to each type of step and weighted proportional to the area and the number of contacts between the directed sides of the objects in general these directed objects are known as festoons being constructed from the so called friendly directed walks while the subset constructed from vicious walks are staircase polygons also known as parallelogram polyominoes additionally we give explicit formulae for various first area moment generating functions that is when the area is summed over all configurations with a given perimeter these results generalize and summarize nearly all known results on the square lattice since such results can be obtained by setting one of the step weights to zero all our results for the triangular lattice are new and hence provide the opportunity to study subtle changes in scaling between lattices in most cases we give our results both in terms of ratios of infinite q series and as continued fractions

exact solution of the staircase and row convex polygon perimeter and area generating function an explicit expression is obtained for the perimeter and area generating function g y z sigma n or 2 sigma m or 1 cn mynzm where cn m is the number of row convex polygons with area m and perimeter n a similar expression is obtained for the area perimeter generating function for staircase polygons both expressions contain q series

osculating and neighbour avoiding polygons on the square lattice we study two simple modifications of self avoiding polygons saps osculating polygons ops are a super set in which we allow the perimeter of the polygon to touch at a vertex neighbour avoiding polygons naps are only allowed to have nearest neighbour vertices provided these are joined by the associated edge and thus form a sub set of saps we use the finite lattice method to count the number of ops and naps on the square lattice we also calculate their radius of gyration and the first area weighted moment analysis of the series confirms exact predictions for the critical exponents and the universality of various amplitude combinations for both cases we have found exact solutions for the number of convex and almost convex polygons

staircase polygons scaling functions and asymmetric compact directed percolation the scaling function for compact directed percolation on a square lattice is investigated for the asymmetric case where two parameters control the critical behaviour a simple representation for the area perimeter generating function for staircase polygons is found which can be recast as a non linear functional equation from this the exact scaling function is extracted in the process the most concise derivations to date are given for the exact low order cluster moments

the size and number of rings on the square lattice the authors have calculated the number of self avoiding polygons on the square lattice to 56 steps and the caliper size to 54 steps analysis of the generating function permits estimates of the connective constant mu 2 638 1585 or 10 6 and the critical exponents alpha 0 500 06 or 0 000 06 and v 0 753 or 0 007 the singularity structure of the polygon generating function is found to be consistent with a correction to the scaling exponent delta 1 5 as predicted by nienhuis 1982 1984 the confluent part however maps into the additive analytic term due to the value of the exponent alpha

the perimeter generating functions of three choice imperfect and one punctured staircase polygons we consider the isotropic perimeter generating functions of three choice imperfect and one punctured staircase polygons whose 8th order linear fuchsian odes are previously known we derive simple relationships between the three generating functions and show that all three generating functions are joint solutions of a common 12th order fuchsian linear ode we find that the 8th order differential operators can each be rewritten as a direct sum of a direct product with operators no larger than 3rd order we give closed form expressions for all the solutions of these operators in terms of 2 f 1 hypergeometric functions with rational and algebraic arguments the solutions of these linear differential operators can in fact be expressed in terms of two modular forms since these 2 f 1 hypergeometric functions can be expressed with two rational or algebraic pullbacks