doubly folded random walks for unentangled polymers a tree percolation model abstract we study the properties of two dimensional unentangled polymer chains in a dense environment of many other cross linked polymer molecules incorporating the excluded volume effect a kinetic random walk model is introduced in which the different possible directions of a step are chosen sequentially in a clockwise direction with a pre assigned probability while the spanning probability and other bulk properties of this walk shows a percolation like transition the detailed structural properties are found to be different a spiraling behavior is observed beyond the percolation threshold we also study two other variants of kinetic random walks

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

an adaptive finite element method for transient compressible flows with moving boundaries a description is given of an adaptive finite element method for the simulation of two dimensional compressible flows with moving boundaries the method is implemented on an unstructured triangular grid and the mesh adaptation is achieved by local grid regeneration using an error estimation procedure coupled to an automatic triangular mesh generator two numerical examples are included to illustrate the numerical performance of the method

examination of the 1 d expansion method from exact enumeration for a self interacting self avoiding walk the 1 d expansion method for polymer chains is examined by comparing these expansions for several thermodynamic and structural quantities with the results of standard series analysis of exact enumeration data the comparisons cover a wide range of spatial dimensions d including non integer ones and are performed for particular values of interaction energy good agreement is generally found for d 4 whereas discrepancies become conspicuous as d decreases to d 2 reasonable values are obtained for the exponents v and gamma in d 2 4 by applying the coherent anomaly method of suzuki 1986 to our 1 d expansions through fifth order in d 1

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

percolation under rotational constraint a finite size scaling study a percolation model spiral percolation in which a rotational constraint is operative is studied by the finite size scaling method the critical percolation probability pc and the critical exponents v beta gamma tau sigma and also the fractal dimension d of the spanning cluster are determined evidence is obtained for a scaling form of the cluster distribution function

e factor of representations of classical groups abstract we let g be the isometry group of a nondegenerate symmetric or skew symmetric form h on a space v over some local or global field k i e g u v h we let lg be the l group of go the identity component of g and r be the natural representation of this group we develop here a theory of l functions and e factors attached to this r in particular we develop a theory of local and global zeta integrals for such g we use very special intertwining operators therein we require an analysis of the proper normalization of such operators with such techniques we determine the correct e factors associated to local representations the issue of the correct local l factor is discussed

the local langlands conjecture for gsp 4 we prove the local langlands conjecture for gsp4 f where f is a nonarchimedean local eld of characteristic zero

geometrical arguments against the alexander orbach conjecture for lattice animals and diffusion limited aggregates geometrical arguments are given which suggest that the alexander orbach conjecture 1982 does not hold for lattice animals in d 2 and for the diffusion limited aggregates for large dimensions

finite volume treatment of dispersion relation preserving and optimized prefactored compact schemes for wave propagation in developing suitable numerical techniques for computational aero acoustics the dispersion relation preserving drp scheme by tam and co workers and the optimized prefactored compact opc scheme by ashcroft and zhang have shown desirable properties of reducing both dissipative and dispersive errors these schemes originally based on the finite difference attempt to optimize the coefficients for better resolution of short waves with respect to the computational grid while maintaining pre determined formal orders of accuracy in the present study finite volume formulations of both schemes are presented to better handle the nonlinearity and complex geometry encountered in many engineering applications linear and nonlinear wave equations with and without viscous dissipation have been adopted as the test problems highlighting the principal characteristics of the schemes and utilizing linear and nonlinear wave equations with different wavelengths as the test cases the performance of these approaches is documented for the linear wave equation there is no major difference between the drp and opc schemes for the nonlinear wave equations the finite volume version of both drp and opc schemes offers substantially better solutions in regions of high gradient or discontinuity

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

unsteady euler solutions for arbitrarily moving bodies and boundaries a methodology for the computation of unsteady euler flows in configurations with moving boundaries is presented the procedure is composed of a moving and adaptive grid management algorithm and a lagrangian eulerian flow solver the domain discretization is handled via unstructured triangular grids the flow scheme uses a generalized version of roe s approximate riemann solver which takes into account the mesh movement while satisfying in an intrinsic way the geometric conservation laws grid flow coupling is obtained through error estimation the methodology is applied to various flow problems showing moving shock waves and moving geometries

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

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

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

branched polymers exact enumeration study of three dimensional lattice animals classified by valence distribution the partial generating function method is applied to the enumeration of lattice animals classified according to their valence distribution this detailed information allows one to study the effects of the number and nature of branch points on the properties of branched polymers several series analysis techniques are applied to the data particular attention is given to the effects of correction to scaling terms

a cut cell non conforming cartesian mesh method for compressible and incompressible flow this paper details a multigrid accelerated cut cell non conforming cartesian mesh methodology for the modelling of inviscid compressible and incompressible flow this is done via a single equation set that describes sub trans and supersonic flows cut cell technology is developed to furnish body fitted meshes with an overlapping mesh as starting point and in a manner which is insensitive to surface definition inconsistencies spatial discretization is effected via an edge based vertex centred finite volume method an alternative dual mesh construction strategy similar to the cell centred method is developed incompressibility is dealt with via an artificial compressibility algorithm and stabilization achieved with artificial dissipation in compressible flow shocks are captured via pressure switch activated upwinding the solution process is accelerated with full approximation storage fas multigrid where coarse meshes are generated automatically via a volume agglomeration methodology this is the first time that the proposed discretization and solution methods are employed to solve a single compressible incompressible equation set on cut cell cartesian meshes the developed technology is validated by numerical experiments the standard discretization and alternative methods were found equivalent in accuracy and computational cost the multigrid implementation achieved decreases in cpu time of up to one order of magnitude copyright 2007 john wiley sons ltd

statistics of lattice animals the authors investigate the large n behaviour of the number of lattice animals with n vertices having alpha cycles per vertex they prove concavity and continuity properties for the corresponding growth constant and in particular show that lattice trees are exponentially scarce in the set of lattice animals they also consider the corresponding generating function and prove a number of theorems which bound it and set other limits on its possible behaviour

cartesian cut cell method for axisymmetric separating body flows a method for the calculation of unsteady axisymmetric compressible flows involving both fixed and separating bodies is presented the method uses a cartesian cut cell mesh approach and high resolution upwind finite volume scheme a stationary background cartesian mesh is generated on the computational domain with complex solid geometries represented by different types of cut cell solid bodies are allowed to move across the mesh using a finite volume scheme modified to deal with moving boundary problems the flow solver employed is a muscl hancock godunov type scheme in conjunction with an approximate riemann solver of the harten lax and van leer type for flow interfaces and an exact riemann solver for a moving piston for fixed or moving solid faces a cell merging technique is used to maintain numerical stability in the presence of arbitrarily small cut cells and to retain strict conservation at moving boundaries the method is applied to a muzzle blast flow and muzzle break problems involving both fixed and separating bodies

a note on coniglio s lemma it is proved that coniglio s 1981 1982 lemma holds for any pair of points in any graph in the low density series expansion of the cut edge weighted pair connectedness

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

multifractality in rotational percolation models abstract in rotationally constrained percolation models a site of a percolation cluster could be occupied more than once from different directions due to the nature of the rotational constraint a state variable s i is assigned to each lattice site whose value corresponds to the number times it has been visited during the growth of a cluster it is proposed here that the percolation transition and the multifractal aspects of infinite percolation clusters under rotational constraint can be studied defining suitable measures in terms of the state variable s i this method does not require to introduce any external agency like an electric current or a random walker in order to explore multifractality as in the case of ordinary percolation the state variable representation also describes the universality class of the percolation models appropriately

theta dichotomy and doubling epsilon factors for spn f in this work we aim to prove the theta dichotomy conjecture for dual pairs of type sp w o v with dim v odd the crux of the proof relies on the analysis of the reducibility of a certain degenerate principal series representation of the group sp w having double the rank of sp w this analysis is carried out by way of a normalized standard intertwining operator which is also used to define an epsilon factor coming from the doubling integral construction of piatetski shapiro and rallis in this paper we also illustrate the connection between the doubling epsilon factor and the theta dichotomy conjecture

1 d expansions for the free energy of weakly embedded site animal models of branched polymers we give the 1 d expansion to order 1 d5 for the limiting reduced free energy of a weakly embedded two variable site animal model of branched polymers hence we are able to derive expansions for the free energy of related one variable models and the growth constant of weakly and strongly embedded trees again to order 1 d5 we argue that the free energy expansions are asymptotically correct for a small range of fugacities only exact results show that animals with a compact hypercubic structure make the overwhelming contribution to the total number of weakly embedded site animals especially for large d this result is not reflected in the free energy 1 d expansions where trees make the dominant contribution we also derive new exact enumeration data for lattice animals using intermediate calculations in the derivation of the 1 d expansions thus we give partition functions for the general d dimensional hypercubic lattice for one to 13 sites

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

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

lattice statistics of branched polymers with specified topologies the authors study the numbers of lattice animals with specified topologies they prove that the growth constant for animals with c cycles and nk vertices of degree k k 3 2d weakly embeddable in a d dimensional hypercubic lattice equal to mu the growth constant for self avoiding walks for some particular topologies they derive upper and lower bounds for the corresponding critical exponents and estimate the values of these exponents by deriving and analysing new exact enumeration data they conjecture that their previous result for trees with b branches that the exponent is gamma b 1 where gamma is the self avoiding walk exponent is also valid in the more general case in which the cyclomatic index c is non zero i e for b or 1 the exponent does not depend on c they show that their results are consistent with renormalisation group arguments that the universality class of branched polymers is independent of cycle and non zero branching fugacities

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

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

a new simulation of branched polymers the scanning method for the simulation of linear chain is extended to general models of branched polymers without loops a branched chain grows in time namely a number of steps from the origin therefore i in contrast to other simulation techniques which are of a relaxation type the chains are statistically independent and the statistical error can reliably be estimated ii the probability of a chain is known and hence the entropy and iii the scanning construction enables one to study geometrical properties which depend on time for self avoiding trees on a square lattice the author obtains the relatively accurate estimates for the static critical exponents nu 0 640 or 0 004 and theta 1 003 or 0 02 and for the connective constant mu 5 1419 or 0 003 the author also obtains critical exponents gamma t approximately 1 26 and nu t approximately 0 83 which characterise the growth in time of the number of bonds and the gyration radius respectively application of the scanning method to more complex branched polymers is discussed

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

monte carlo test of dimensional reduction for branched polymers in three dimensions by a monte carlo simulation of a grand canonical ensemble for a lattice tree model of branched polymers in three dimensions the author obtains the estimates theta 1 501 or 0 043 nu 0 495 or 0 013 for the critical exponents these estimates strongly support the exact conjecture theta 3 2 nu 1 2 by parisi and sourlas 1981 obtained using dimensional reduction the attrition constant mu is also estimated

lattice trees with a restricted number of branch points the authors investigate the asymptotic behaviour of the number of trees having n vertices weakly embeddable in the d dimensional hypercubic lattice with restrictions on the number n of vertices with degree greater than two they show rigorously that if n o n log n the growth constant is equal to the corresponding quantity for self avoiding walks for n increasing linearly with n they show that the growth constant still exists and present arguments indicating that it is strictly greater than that for self avoiding walks

universality in branched polymers on d dimensional hypercubic lattices the authors have derived series for weakly and strongly embeddable trees in d dimensional simple hypercubic lattices for arbitrary integral d for d 2 3 9 they present series evidence that such trees are in the same universality class as lattice animals in addition they have derived expansions in inverse powers of sigma 2d 1 for the growth parameters for bond and site trees and compare these with the corresponding results for animals

cartesian euler method for arbitrary aircraft configurations a three dimensional euler method is described for steady flow calculations about arbitrary aircraft configurations the method uses multigrid calculations with equally meshed cartesian not necessarily rectangular grids and local refinement the use of local computational grids that are not aligned to the body surface or even possibly not aligned to one another removes the need for complicated grid generation but introduces the need for sophisticated boundary condition implementation

the collapse transition for the lattice trees the authors consider a lattice tree model for the collapse of dilute branched polymers in the good solvent regime in which the collapse is driven by a near neighbour contact fugacity they describe some rigorous results including bounds for the temperature dependence of the reduced limiting free energy and compare these results with numerical estimates derived from exact enumeration data from the specific heat they estimate the collapse temperature tc and the cross over exponent phi 0 in two and three dimensions they find phi 0 0 60 or 0 03 d 2 and phi 0 0 82 or 0 03 d 3 finally they speculate on a possible roughening transition which may occur at a temperature tr tc

adaptive remeshing for three dimensional compressible flow computations an adaptive mesh procedure for computing steady state solutions of the compressible euler equations in three dimensions is presented the method is an extension of previous work in two dimensions the approach requires the coupling of a surface triangulator an automatic tetrahedral mesh generator a finite element flow solver and an error estimation procedure an example involving flow at high mach number is included to demonstrate the numerical performance of the proposed approach the example shows that the use of this form of adaptivity in three dimensions offers the potential of even greater computational savings than those attained in the corresponding two dimensional implementation

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

cluster renormalisation study of site lattice animals in two and three dimensions the author studies site lattice animals using the cluster renormalisation cr approach previously applied to bond animals the exponent v characterising the asymptotic behaviour of the mean square radius of gyration of site animals and the fractal dimension df 1 v are determined in two and three dimensions he finds v 0 649 or 0 009 and v approximately 0 51 in two and three dimensions respectively in addition he studies restricted valence site animals on a triangular lattice using a single parameter cr approach

branched polymers with a prescribed number of cycles the authors consider the number of lattice animals with exactly c cycles for weakly embeddable bond clusters they show rigorously that the growth constant is independent of c and they derive upper and lower bounds on the critical exponents for each value of c they use series analysis methods to estimate the critical exponents for c 1 and 2 in two and three dimensions and find that the critical exponent does depend on the number of cycles evidence from series analysis results and from expansions in inverse powers of the dimension suggests that in the case of strong embeddings i e site clusters the growth constant is independent of c and the corresponding values of the critical exponents are identical to the values for weak embeddings i e bond clusters they discuss the relationship of these results to the field theory prediction that the critical exponent is independent of cycle fugacity

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

on the percolation threshold for a d dimensional simple hypercubic lattice the authors use 1 sigma expressions sigma 2d 1 to investigate systematically a number of conjectures and approximations for the bond and site percolation thresholds of a d dimensional simple hypercubic lattice the technique provides convincing evidence that none of the conjectures are correct and gives a quantitative understanding of the quality of the approximations

theta dichotomy for unitary groups some recent work of gross and prasad 14 suggests that the root numbers attached to certain symplectic representations of the weil deligne group of a local field f control certain branching rules for representations of orthogonal groups over f on a global level this local phenomenon should have implications for the structure of the value at the center of symmetry for certain l functions of arithmetic interest 12 this conjectural picture is based to some extent on the now classic work of tunnell 47 and waldspurger 49 as well as on the case of the triple product l function 13 34 17 in all of these examples the local root number detects the existence of a certain type of invariant linear functional it is possible to set up analogous conjectures for the isometry groups of hermitian or quaternion hermitian forms it turns out that root numbers also play a role in the local and global theta correspondence roughly speaking certain local root numbers should control the occurrence of representations in the local theta correspondence between groups of the same size e g for dual pairs of the form sp n o 2n 1 and u n u n as will be seen the theta correspondence for such pairs is connected with a certain induced representation in s at the point s 0 on the unitary axis by contrast the correspondence for the pairs sp n o 2n and sp n o 2n 2 discussed by prasad 35 is connected to the behavior of a similar induced representation at the points 12 in which case no epsilon factor arises in this paper we consider the local theta correspondence for unitary groups in the non archimedean case let f be a non archimedean local field of characteristic not equal to 2 and let e be a quadratic extension of f for further notation see the notation section at the end of this introduction let v and w be e vector spaces of dimensions m and n equipped respectively with a hermitian form v v e and a skew hermitian form w w e then the isometry groups g v and g w of the spaces v and w form a dual reductive pair in the symplectic group sp w where w v e w viewed as an f vector space of dimension 2mn and equipped with the symplectic form

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

critical exponents for c animals with nearest neighbour interaction the authors study two lattice models a c animals with an interaction energy alpha between nearest neighbour pairs of vertices b c animals as in a but with an additional interaction energy omega between the vertices of the animal and an adsorption surface by assuming that the partition functions satisfy an c alpha approximately n theta c alpha lambda c alpha n and an c alpha omega approximately n theta c alpha omega lambda c alpha omega n as n to infinity with c fixed they show that theta c alpha theta 0 alpha c infinity alpha infinity and theta c alpha omega theta 0 alpha omega c infinity alpha omega infinity where theta 0 alpha and theta 0 alpha omega are the corresponding exponents for trees

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

critical exponents for lattice animals with fixed cyclomatic index the authors derive an inequality between the number of trees and the number of lattice animals with exactly c cycles an c for all positive c if they assume that an c approximately n theta c lambda nc n to infinity c fixed they use this to show that theta c theta o c where theta 0 is the corresponding exponent for trees

the doubling integral for the principal series of spn f and sp n f abstract in a previous article we explicitly computed the doubling integral for constituents of the unramified principal series for sp 2 f and sp 2 f those computations relied on some intermediate constructions related to the theory of local theta liftings that make generalizing the computations to higher rank groups or more ramified representations quite difficult this article attempts to compute the doubling integral for arbitrary constituents of the possibly ramified principal series representations for arbitrary rank symplectic and metaplectic groups to do this we circumvent the intermediate computations from the previous article avoid constructions relying on the theory of local theta lifts and interpret the integration over various unipotent subgroups as standard intertwining operators we would also like to thank the referees for their careful reading of this manuscript as well as their suggestions for its improvement

effect of loops on spiral lattice animal statistics the authors have shown by exact enumeration on a square lattice that spiral lattice site animals and spiral lattice site trees belong to different universality classes i e loops have a non trivial effect on spiral lattice animal statistics this result is contrary to what is known in the cases of undirected and directed lattice animals some special features of spiral lattice animals have been pointed out in this context

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

l functions and theta correspondence for classical groups using the doubling method of piatetski shapiro and rallis we develop a theory of local factors of representations of classical groups and apply it to give a necessary and sufficient condition for nonvanishing of global theta liftings in terms of analytic properties of the l functions and local theta correspondence

a rigorous bound on the critical exponent for the number of lattice trees animals and polygons the number ofn site lattice trees up to translation is believed to behave asymptotically ascn 0 n where is a critical exponent dependent only on the dimensiond of the lattice we present a rigorous proof that d 1 d for anyd 2 the method also applies to lattice animals site animals and two dimensional self avoiding polygons we also prove that v whend 2 wherev is the exponent for the radius of gyration

a tree based scaling exponent for random cluster models for models of random trees in the hypercubic lattice or more general random clusters to which tree structure may be added the authors introduce a novel scaling exponent for the particular model of uniform random spanning trees in two and three dimensions its value is estimated by monte carlo simulation

simulations of lattice animals and trees the scaling behaviour of randomly branched polymers in a good solvent is studied in two to nine dimensions using as microscopic models lattice animals and lattice trees on simple hypercubic lattices as a stochastic sampling method we use a biased sequential sampling algorithm with re sampling similar to the pruned enriched rosenbluth method perm used extensively for linear polymers essentially we start simulating percolation clusters either site or bond re weigh them according to the animal tree ensemble and prune or branch the further growth according to a heuristic fitness function in contrast to previous applications of perm this fitness function is not the weight with which the actual configuration would contribute to the partition sum but is closely related to it we obtain high statistics of animals with up to several thousand sites in all dimension 2 d 9 in addition to the partition sum number of different animals we estimate gyration radii and numbers of perimeter sites in all dimensions we verify the parisi sourlas prediction and we verify all exactly known critical exponents in dimensions 2 3 4 and 8 in addition we present the hitherto most precise estimates for growth constants in d 3 for clusters with one site attached to an attractive surface we verify for d 3 the superuniversality of the cross over exponent at the adsorption transition predicted by janssen and lyssy but not for d 2 there we find 0 480 4 instead of the conjectured 1 2 finally we discuss the collapse of animals and trees arguing that our present version of the algorithm is also efficient for some of the models studied in this context but showing that it is not very efficient for the classical model for collapsing animals

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

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

algebraic theory of dicritical divisors the topological theory of dicritical divisors is algebracized which makes it valid for nonzero characteristic as well as mixed characteristic

on c animals interacting with a surface the authors examine the interaction of weakly embedded c animals with an adsorption surface and show that such animals have the same reduced free energy as that for weakly embedded trees by assuming the existence of theta 0 omega the critical exponent for trees they show that theta c omega exists and theta c omega theta 0 omega c for all omega

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

a new monte carlo simulation for two models of self avoiding lattice trees in two dimensions by means of a new monte carlo sampling of a grand canonical ensemble we verify universality for the critical exponents and of two models of lattice trees constrained to be self avoiding on sites or on bonds the attrition constants are also obtained this algorithm a generalization of that recently proposed by berretti and sokal for random walks appears to optimize the critical slowing down in the scaling region systematic and statistical errors are carefully estimated

a two dimensional conservation laws scheme for compressible flows with moving boundaries the generalized riemann problem grp scheme for the hydrodynamic conservation laws is extended to a two dimensional moving boundary tracking mbt configuration aimed at treating time dependent compressible flows with moving impermeable boundary surfaces a strang type operator splitting is employed for the integration of the conservation laws the boundary motion is also split into cartesian components that are taken with the respective operator split integration phases the conservation laws in boundary cells are integrated by a finite volume scheme that accounts for changing mesh geometry cell volume and side area through which mass momentum and energy fluxes are taken the central feature of the scheme is the algorithms for evaluating the changing mesh geometry at boundary cells these algorithms are based on a decomposition rule for evaluating polygon intersection area which has been used extensively for rezoning in hydrocodes over the past decades the decomposition rule is combined with the cartesian splitting of the boundary motion producing algorithms for the integration of conservation laws in boundary cells that are both consistent and simple consistency is taken to mean that the scheme produces an exact solution in the case of a uniform motion of the fluid and the boundary common velocity an illustrative example of shock lifting of a light weight cylinder is presented

unsteady euler algorithm with unstructured dynamic mesh for complex aircraft aerodynamic analysis an euler solution algorithm is presented for unsteady aerodynamic analysis of complex aircr aft configurations the flow solver involves a multistage runge kutta time stepping scheme that uses a finite volume spatial discretization on an unstructured grid made up of tetrahedra a significant contribution of the research is the development and implementation of a moving mesh algorithm that is employed for problems involving static or dynamic deformation of the aircraft the mesh algorithm is a general procedure that can treat realistic motions and deformations of complex aircraft configurations steady and unsteady results are presented for a supersonic fighter configuration to demonstrate applications of the euler solver and dynamic mesh algorithm the unsteady flow results were obtained for the aircraft oscillating harmonically in a complete vehicle bending mode effects of angle of attack and reduced frequency on instantaneous pressures and force responses were investigated the paper presents descriptions of the euler solver and dynamic mesh algorithm along with results that assess the capability

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