umbilical torus bifurcations in hamiltonian systems we consider perturbations of integrable hamiltonian systems in the neighbourhood of normally umbilic invariant tori these lower dimensional tori do not satisfy the usual non degeneracy conditions that would yield persistence by an adaption of kam theory and there are indeed regions in parameter space with no surviving torus we assume appropriate transversality conditions to hold so that the tori in the unperturbed system bifurcate according to a generalised umbilical catastrophe combining techniques of kam theory and singularity theory we show that such bifurcation scenarios of invariant tori survive the perturbation on large cantor sets applications to gyrostat dynamics are pointed out

the thermodynamic limit for long range random systems long range spin systems with random interactions are considered a simple argument is presented showing that the thermodynamic limit of the free energy exists and depends neither on the specific random configuration nor on the sample shape provided there is no external field the argument is valid for both classical and quantum spin systems and can be applied to a spins randomly distributed on a lattice and interacting via dipolar interactions and b spin systems with potentials of the formj x 1 x 2 x 1 x 2 d where thej x 1 x 2 are independent random variables with mean zero d is the dimension and 1 2 the key to the proof is a multidimensional subadditive ergodic theorem as a corollary we show that for random ferromagnets the correlation length is a nonrandom quantity

localization of a random copolymer at an interface an untethered self avoiding walk model we consider two related n step self avoiding walk models of a copolymer at an interface between two bulk phases in one case the walk is confined to start in the interface while in the other this condition is relaxed we prove that both models have the same limiting free energy in the n limit and hence that their phase diagrams are identical we also show that the limits n and certain energy parameters going to plus or minus infinity can be interchanged these latter results are interesting from the viewpoint of numerical studies

extent of self averaging in the statistical mechanics of finite random copolymers we investigate the extent of thermodynamic self averaging in a coloured self avoiding walk model of finite random copolymer adsorption we derive a bound on the extent of self averaging as a function of the length of the self avoiding walk

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

interacting self avoiding walks and polygons in three dimensions self interacting walks and polygons on the simple cubic lattice undergo a collapse transition at the point we consider self avoiding walks and polygons with an additional interaction between pairs of vertices which are unit distance apart but not joined by an edge of the walk or polygon we prove that these walks and polygons have the same limiting free energy if the interactions between nearest neighbour vertices are repulsive the attractive interaction regime is investigated using monte carlo methods and we find evidence that the limiting free energies are also equal here in particular this means that these models have the same point in the asymptotic limit the dimensions and shapes of walks and polygons are also examined as a function of the interaction strength

self averaging in random self interacting polygons we prove that a lattice polygon model of a self interacting random ring copolymer is thermodynamically self averaging the proof is quite general and applies to any two body potential linear in the numbers of contacts of different types

self averaging in finite random copolymers we investigate how martingale techniques can be used to derive information on the extent of self averaging of the free energy for some lattice models of finite random copolymers

a self avoiding walk model of random copolymer adsorption there is an error in the proof of lemma 3 4 which can be remedied by using 3 22 and 3 13 directly the proof of lemma 3 2 can also be simplified please see pdf for details

simple strong glass forming models mean field solution with activation we introduce simple models inspired by previous models for froths and covalent glasses with trivial equilibrium properties but dynamical behaviour characteristic of strong glass forming systems these models are also a generalization of backgammon or urn models to a non constant number of particles where entropic barriers are replaced by energy barriers allowing for the existence of activated processes we formulate a mean field version of the models which keeps most of the features of the finite dimensional ones and solve analytically the out of equilibrium dynamics in the low temperature regime where activation plays an essential role

adsorption of a random heteropolymer at a potential well revisited location of transition point and design of sequences the adsorption of an ideal heteropolymer loop at a potential point well is investigated within the framework of standard random matrix theory on the basis of a semi analytical semi numerical approach a histogram of transition points for the ensemble of quenched heteropolymer structures with bimodal symmetric distribution of types of chain links is constructed it is shown that the sequences having transition points in the tail of the histogram display correlations between nearest neighbour monomers

a numerical study of self averaging in adsorption of random copolymers and random surfaces numerical studies involving random copolymers and random surfaces assume self averaging of thermodynamic and metric properties of the systems to calculate different properties for the problem of adsorption of a random copolymer rigorous proofs regarding self averaging of some properties such as free energy in the thermodynamic limit n exist this says little about the extent of self averaging for finite size systems used in numerical studies for the problem of adsorption of a homopolymer on a random surface no analytical proofs regarding self averaging exist in this work assumptions of self averaging of thermodynamic and metric properties of a self avoiding walk model of random copolymer adsorption are tested via multiple markov chain monte carlo method numerical evidence is provided in support of self averaging of energy heat capacity and the z component of the self avoiding walk in different temperature intervals self averaging in energy of a homopolymer interacting with a random surface is also examined

density and correlation functions of vortex and saddle points in open billiard systems we present microwave measurements for the density and spatial correlation of current critical points in an open billiard system and compare them with new and previous predictions of the random wave model rwm in particular due to an improvement of the experimental setup we determine experimentally the spatial correlation of saddle points of the current field an asymptotic expression for the vortex saddle and saddle saddle correlation functions based on the rwm is derived with experiment and theory agreeing well we also derive an expression for the density of saddle points in the presence of a straight boundary with general mixed boundary conditions in the rwm and compare with experimental measurements of the vortex and saddle density in the vicinity of a straight wall satisfying dirichlet conditions

signed zeros of gaussian vector fields density correlation functions and curvature we calculate correlation functions of the signed density of zeros of gaussian distributed vector fields we are able to express correlation functions of arbitrary order through the curvature tensor of a certain abstract riemann cartan or riemannian manifold as an application we discuss one and two point functions the zeros of a two dimensional gaussian vector field model the distribution of topological defects in the high temperature phase of two dimensional systems with orientational degrees of freedom such as superfluid films thin superconductors and liquid crystals

screening of charged singularities of random fields many types of point singularity have a topological index or charge associated with them for example the phase of a complex field depending on two variables can either increase or decrease on making a clockwise circuit around a simple zero enabling the zeros to be assigned charges of 1 in random fields we can define a correlation function for the charge weighted density of singularities for many types of topologically charged singularity this correlation function satisfies an identity which shows that the singularities screen each other perfectly a positive singularity is surrounded by an excess of concentration of negatives which exactly cancel its charge and vice versa this paper gives a simple and widely applicable derivation of this result a counterexample where screening is incomplete is also exhibited

on the number of critical points of the period in the study of subharmonic bifurcations from a planar hamiltonian system it is often assumed that the period of a one parameter family of periodic orbits of the unperturbed system is a monotone function of the parameter see e g hale and taboas s chow and hale es however this hypothesis is in many instances difficult to verify for example it is well known that every orbit in the first quadrant of the vollterra lotka equation is periodic it is only in recent years that the monotonicity of the periods of these periodic orbits has been shown hsu 9 and waldvogel 151 such problems also occur in the study of bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation u f u o uer 1 1 on some interval in 14 smoller and wasserman showed that if f u u u u u cc

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

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

on the quasi periodic d fold degenerate bifurcation abstract this paper analyses the d fold degenerate bifurcation of invariant quasi periodic tori of normal dimension one it is shown that for integrable families of diffeomorphisms unfolding a d fold degenerate invariant torus versally invariant diophantine tori in the family persist under a small non integrable perturbation as a consequence a subset of positive measure with respect to an interpolating smooth manifold of the bifurcation set persists as well the proof is a consequence of the translated torus theorem of russmann and herman

parabolic resonances in 3 degree of freedom near integrable hamiltonian systems abstract perturbing an integrable 3 degree of freedom d o f hamiltonian system containing a normally parabolic 2 torus which is m resonant m 1 or 2 creates a parabolic m resonance m pr prs of different types are either persistent or of low co dimension hence they appear robustly in many applications energy momenta bifurcation diagram is constructed as a tool for studying the global structure of 3 d o f near integrable systems a link between the diagram shape pr and the resonance structure is found the differences between the dynamics appearing in 2 and 3 d o f systems exhibiting prs are studied analytically and numerically the numerical study demonstrates that prs are an unavoidable source of large and fast instabilities in typical 3 d o f systems

self averaging in models of random copolymer collapse we give a set of conditions under which a system is thermodynamically self averaging and show that several lattice models of interacting copolymers satisfy these conditions we prove this result for a general potential which is linear in the numbers of various types of contacts and show that this includes two potentials which have previously been used in models of random interacting linear copolymers

the statistical mechanics of random copolymers random copolymers are polymers with two or more types of monomer where the monomer sequence is determined by some random process once determined the sequence is fixed so random copolymers are an example of a system with quenched randomness we review the statistical mechanics of random copolymers focusing on self avoiding walk models where there are two types of monomers a and b which are randomly distributed along the polymer chain theoretical approximate and numerical results are reviewed for models of the random copolymer adsorption localization and collapse phase transitions we concentrate on what is known about the existence of phase transitions the morita approximation and results about self averaging we also discuss in less detail the replica trick and numerical methods including monte carlo methods exact enumeration and transfer matrix methods important open problems are identified throughout and highlighted in the conclusions

compensations in small divisor problems a general direct method alternative to kam theory apt to deal with small divisor problems in the real analytic category is presented and tested on several small divisor problems including the construction of maximal quasi periodic solutions for nearly integrable non degenerate hamiltonian or lagrangian systems and the construction of lower dimensional resonant tori for nearly integrable hamiltonian systems the method is based on an explicit graph theoretical representation of the formal power series solutions which allows to prove compensations among the monomials forming such representation

normal linear stability of quasi periodic tori abstract we consider families of dynamical systems having invariant tori that carry quasi periodic motions our interest is the persistence of such tori under small nearly integrable perturbations this persistence problem is studied in the dissipative the hamiltonian and the reversible setting as part of a more general kam theory for classes of structure preserving dynamical systems this concerns the parametrized kam theory as initiated by moser j k moser on the theory of quasiperiodic motions siam rev 8 2 1966 145 172 j k moser convergent series expansions for quasi periodic motions math ann 169 1967 136 176 and further developed in g b huitema unfoldings of quasi periodic tori phd thesis university of groningen 1988 h w broer g b huitema f takens unfoldings of quasi periodic tori mem amer math soc 83 421 1990 1 82 h w broer g b huitema unfoldings of quasi periodic tori in reversible systems j dynam differential equations 7 1 1995 191 212 the corresponding nondegeneracy condition involves certain trans versality conditions on the normal linear leading part at the invariant tori we show that as a consequence a cantor family of diophantine tori with positive hausdorff measure is persistent under nearly integrable perturbations this result extends the above references since presently the case of multiple floquet exponents is included our leading example is the normal 1 1 resonance which occurs a lot in applications both hamiltonian and reversible as an illustration of this we briefly describe the lagrange top coupled to an oscillator

the quasi periodic centre saddle bifurcation abstract nearly integrable families of hamiltonian systems are considered in the neighbourhood of normally parabolic invariant tori in the integrable case such tori bifurcate into normally elliptic and normally hyperbolic invariant tori with a kam theoretic approach it is shown that both the normally parabolic tori and the bifurcation scenario survive a non integrable perturbation parametrised by pertinent large cantor sets these results are applied to rigid body dynamics