adsorbing motzkin paths the adsorption of a motzkin path with lifted endpoints onto an adsorbing line is examined as a model of polymer adsorption the partition function of this model is determined exactly and the free energy is derived by examining the partition function the location of the critical point in the model is given exactly in one case and as the solution of a nonlinear equation in another case the phase diagram of the model is examined and i show that the transition is in general a first order adsorption transition except in cases where one or both endpoints of the path are restricted to be close to the adsorbing line

adsorbing and collapsing directed animals a model of a self interacting directed animal which also interacts with a solid wall is studied as a model of a directed branched polymer which can undergo both a collapse and an adsorption transition the directed animal is confined to a 45 wedge and it interacts with one of the walls of this wedge the existence of a thermodynamic limit in this model shown and the presence of an adsorption transition is demonstrated by using constructive techniques by comparing this model to a process of directed percolation we show that there is also a collapse or transition in this model we examine directed percolation in a wedge to show that there is a collapse phase present for arbitrary large values of the adsorption activity the generating function of adsorbing directed animals in a half space is found next from which we find the tricritical exponents associated with the adsorption transition a full solution for a collapsing directed animal seems intractible so instead we examine the collapse transition of a model of column convex directed animals with a contact activity next

pulled motzkin paths in this paper the models of pulled dyck paths in janse van rensburg 2010 j phys a math theor 43 215001 are generalized to pulled motzkin path models the generating functions of pulled motzkin paths are determined in terms of series over trinomial coefficients and the elastic response of a motzkin path pulled at its endpoint see orlandini and whittington 2004 j phys a math gen 37 5305 14 is shown to be r f 0 for forces pushing the endpoint toward the adsorbing line and r f f 1 2cosh f 2sinh f f as f for forces pulling the path away from the x axis in addition the elastic response of a motzkin path pulled at its midpoint is shown to be r f 0 for forces pushing the midpoint toward the adsorbing line and r f f 1 2cosh f 2 sinh f 2 2f as f for forces pulling the path away from the x axis formal combinatorial identities arising from pulled motzkin path models are also presented these identities are the generalization of combinatorial identities obtained in directed paths models to their natural trinomial counterparts

periodic orbits near bifurcations of codimension two classical mechanics semiclassics and stokes transitions we investigate classical and semiclassical aspects of codimension two bifurcations of periodic orbits in hamiltonian systems a classification of these bifurcations in autonomous systems with two degrees of freedom or time periodic systems with one degree of freedom is presented we derive uniform approximations to be used in semiclassical trace formulae and determine also certain global bifurcations in conjunction with stokes transitions that become important in the ensuing diffraction catastrophe integrals

pulled directed lattice paths the generating functions of models of directed walks pulled by an external force f are determined using the kernel method these paths are models of linear polymers subject to an external force the generating function is related to the generating function of ballot paths and has an unexpected and non physical singularity for the model of dyck paths pulled at its central vertex in each model the force extension curve is determined exactly and has a sigmoid shape partially given by ctanh cf if f is the applied force and where c and c are the model dependent constants this result is consistent with previous results for models of pulled directed paths and also with data obtained from the numerical simulation of self avoiding walk models of linear polymers

a collapse transition in a directed walk model the authors consider a directed walk model of linear polymers in dilute solution with an energy associated with the number of near neighbour contacts in the walk for this model they can derive an exact expression for the generating function in two variables conjugate to the number of steps and the number of contacts they discuss the analytic structure of this generating function and identify the transition corresponding to collapse

on quantisation using periodic classical orbits two approaches to semiclassical quantisation of integrable systems using periodic classical orbits are considered they both lead to approximate formulae for the density of states function a delta function at each energy level the first due to gutzwiller 1971 involves a sum over isolated stable periodic orbits of the system and leads to the harmonic approximation to the eigenvalues the second due to berry and tabor 1976 involves a sum over families of periodic orbits and leads to the ebk torus approximation to the eigenvalues here the author extracts a modified form of the gutzwiller series from the berry tabor series by using a uniform approximation and hence show that the complete spectrum involves both these series the analysis demonstrates that genuine semiclassical quantisation rules for generic systems using periodic orbits will involve uniform approximation which more closely reflects the underlying classical structure than do the existing stationary phase approximations

semiclassical trace formulas for pitchfork bifurcation sequences in non integrable hamiltonian systems with mixed phase space and discrete symmetries sequences of pitchfork bifurcations of periodic orbits pave the way from integrability to chaos in extending the semiclassical trace formula for the spectral density we develop a uniform approximation for the combined contribution of pitchfork bifurcation pairs for a two dimensional double well potential and the familiar h enon heiles potential we obtain very good agreement with exact quantum mechanical calculations we also consider the integrable limit of the scenario which corresponds to the bifurcation of a torus from an isolated periodic orbit for the separable version of the h enon heiles system we give an analytical uniform trace formula which also yields the correct harmonic oscillator su 2 limit at low energies and obtain excellent agreement with the slightly coarse grained quantum mechanical density of states

the effect of attractive monomer monomer interactions on adsorption of a polymer chain the authors have used transfer matrix techniques to study a model of polymer adsorption and collapse they consider the conventional model of a self avoiding walk with attractive nearest neighbour interactions on the square lattice and introduce adsorbing boundaries the numerical results are consistent with the existence of a multicritical point where the adsorption and collapse transitions coincide and suggest that the adsorption transition of a collapsed polymer chain in first order there is also clear evidence that the collapse transition lowers the adsorption temperature estimates of the surface scaling dimensions at the ordinary theta transition support the conjecture of seno and stella 1988 that the surface critical behaviour at the theta and theta points belong to different universality classes

uniform approximation for bifurcations of periodic orbits with high repetition numbers the semiclassical contribution of a periodic orbit to the quantum density of states diverges when the orbit bifurcates in this case one has to apply approximations which are uniformly valid both in and a parameter which describes the distance to the bifurcation the form of the approximation depends on the repetition number m of the orbit that bifurcates in a two dimensional system the approximations are different for m 1 up to m 5 and for they have the same form as for m 5 in this article we consider the case which occurs first when an integrable system is perturbed a uniform approximation for the contribution to the spectral density is derived which in the limit of large reduces to a sum of semiclassical contributions of isolated periodic orbits

tunnelling in near integrable systems tunnelling rates are qualitatively different for integrable and near integrable systems here a uniform result is derived which interpolates between the two regimes and is applied successfully to two dimensional double well and pendulum potentials when the underlying symmetry is reflection in a single coordinate the splitting remains positive after perturbation but for potentials whose underlying symmetry is reflection through the origin the splitting is predicted to oscillate as a function of a perturbation parameter with regularly spaced zeros where tunnelling switches off

string equations in whitham hierarchies τ functions and virasoro constraints a scheme for solving whitham hierarchies satisfying a special class of string equations is presented the function of the corresponding solutions is obtained and the differential expressions of the underlying virasoro constraints are characterized illustrative examples of exact solutions of whitham hierarchies are derived and applications to conformal maps dynamics are indicated

a directed walk model of a long chain polymer in a slit with attractive walls we present the exact solutions of various directed walk models of polymers confined to a slit and interacting with the walls of the slit via an attractive potential we consider three geometric constraints on the ends of the polymer and concentrate on the long chain limit apart from the general interest in the effect of geometrical confinement this can be viewed as a two dimensional model of steric stabilization and sensitized flocculation of colloidal dispersions we demonstrate that the large width limit admits a phase diagram that is markedly different from the one found in a half plane geometry even when the polymer is constrained to be fixed at both ends on one wall we are not able to find a closed form solution for the free energy for finite width at all values of the interaction parameters but we can calculate the asymptotic behaviour for large widths everywhere in the phase plane this allows us to find the force between the walls induced by the polymer and hence the regions of the plane where either steric stabilization or sensitized flocculation would occur

the effect of pitchfork bifurcations on the spectral statistics of hamiltonian systems we present a quantitative semiclassical treatment of the effects of bifurcations on the spectral rigidity and the spectral form factor of a hamiltonian quantum system defined by two coupled quartic oscillators which on the classical level exhibits mixed phase space dynamics we show that the signature of a pitchfork bifurcation is two fold beside the known effect of an enhanced periodic orbit contribution due to its peculiar hbar dependence at the bifurcation we demonstrate that the orbit pair born at the bifurcation gives rise to distinct deviations from universality slightly above the bifurcation this requires a semiclassical treatment beyond the so called diagonal approximation our semiclassical predictions for both the coarse grained density of states and the spectral rigidity are in excellent agreement with corresponding quantum mechanical results

euler s exemplum memorabile inductionis fallacis and q trinomial coefficients the trinomial coefficients are defined centrally by zji m m 2xi 1 x x i m euler observed that for i fm fm 1 where fm is the mth fibonacci number the assertion is false for m 7 we prove general identities one of which reduces to euler s as sertion for m observation among other things we are led to finite versions of dissections of the rogers ramanujan identities into even and odd parts department of mathematics the pennsylvania state university university park pennsylvania 16802 this content downloaded from 207 46 13 33 on wed 12 oct 2016 04 56 08 utc all use subject to http about jstor org terms

universal twinkling exponents for spectral fluctuations associated with mixed chaology for systems that are neither fully integrable nor fully chaotic bifurcations of periodic orbits give rise to semiclassically emergent singularities in the fluctuating part n fl of the energy level counting function the bifurcations dominate the spectral moments m m n fl 2 m in the limit 0 we argue that m m const vm and calculate the twinkling exponents v m as the result of a competition between bifurcations with different codimensions and repetition numbers

new results for directed vesicles and chains near an attractive wall in this paper we present new exact results for single fully directed walks and fully directed vesicles near an attractive wall this involves a novel method of solution for these types of problems the major advantage of this method is that it unlike many other single walker methods generalizes to an arbitrary number of walkers the method of solution involves solving a set of partial difference equations with a bethe ansatz the solution is expressed as a constant term formula which evaluates to sums of products of binomial coefficients the vesicle critical temperature is found at which a binding transition takes place and the asymptotic forms of the associated partition functions are found to have three different entropic exponents depending on whether the temperature is above below or at its critical value the expected number of monomers adsorbed onto the surface is found to become proportional to the vesicle length at temperatures below critical scaling functions near the critical point are determined

resonant periodic orbits and the semiclassical energy spectrum the semiclassical density of states depends according to the periodic orbit sum formula on the linear stability of the orbits this means however that contributions from the marginally stable or resonant orbits which necessarily accompany stable ones diverge unphysically the remedy for a system of two degrees of freedom is found to lie in the classical non linear normal forms for periodic orbits which describe how satellite periodic orbits coalesce with the central one as resonance is approached in to 0 through these forms the resonant contributions are expressed as diffraction integrals the first few being diffraction catastrophes uniformly valid in in and h cross and finite even for in to 0 provided h cross 0 an extension is proposed to incorporate jointly multiple resonances found in repetitions of orbits

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

semiclassical trace formulae for systems with non abelian symmetry the authors derive generalizations of the trace formula of gutzwiller 1967 and balian and bloch 1970 that are valid in the presence of a non abelian continuous symmetry the usual trace formula must be modified in such cases because periodic orbits occur in continuous families whereas the usual trace formula requires that the periodic orbits be isolated at a given energy these calculations extend the results of a previous paper in which they considered abelian continuous symmetries the most important application of the results is to systems with full three dimensional rotational symmetry and they give this case special consideration

basic analytic combinatorics of directed lattice paths this paper develops a unified enumerative and asymptotic theory of directed two dimensional lattice paths in half planes and quarter planes the lattice paths are specified by a finite set of rules that are both time and space homogeneous and have a privileged direction of increase they are then essentially one dimensional objects the theory relies on a specific kernel method that provides an important decomposition of the algebraic generating functions involved as well as on a generic study of singularities of an associated algebraic curve consequences are precise computable estimates for the number of lattice paths of a given length under various constraints bridges excursions meanders as well as a characterization of the limit laws associated to several basic parameters of paths

bifurcations of periodic orbits and uniform approximations we derive uniform approximations for contributions to gutzwiller s periodic orbit sum for the spectral density which are valid close to bifurcations of periodic orbits in systems with mixed phase space there orbits lie close together and give collective contributions while the individual contributions of gutzwiller s type would diverge at the bifurcation new results for the tangent the period doubling and the period tripling bifurcation are given they are obtained by going beyond the local approximation and including higher order terms in the normal form of the action the uniform approximations obtained are tested on the kicked top and are found to be in excellent agreement with exact quantum results

motzkin path models of long chain polymers in slits we consider motzkin path models for polymers confined to a slit the path interacts with each of the two confining lines and we define parameters a and b to characterize the strengths of the interactions with the two lines we consider the cases where i each vertex in a confining line contributes to the energy and ii where each edge in a confining line contributes to the energy for the vertex and edge versions of motzkin paths we can find the generating functions and give rigorous explicit expressions for the free energy at some special points in the a b plane and asymptotically i e for large slit widths elsewhere we find regions where the force between the lines is long range and repulsive short range and repulsive and short range and attractive our results indicate that the general form of the phase diagram is model independent and similar to previous results for a dyck path model although the details do depend on the underlying configurational model we also contrast the method used here to find the generating function with the transfer matrix and heap methods

near integrable systems resonances and semiclassical trace formulas trace formulas relate the quantum density of states to the properties of the periodic orbits of the underlying classical system the resulting expressions depend critically on the nature of the dynamics and whether the orbits are stable or unstable several open questions exist for the class of classical systems that are near integrability the most important consequence of a generic perturbation to an integrable system is the creation of resonances we derive generalized expressions appropriate for resonances and apply them to a system that can be taken as a paradigm for the transition from regular to chaotic dynamics s1063 651x 96 06506 3