random copolymers and the morita approximation polymer adsorption and polymer localization we analyse directed walk models of random copolymer adsorption and localization ideally we would like t os olve thequenched problem but it appears to be intractable even for simple directed models the annealed approximation is solvable but is inadequate in the strong interaction regime fo rt he adsorption problem and gives a qualitatively incorrect phase diagram for the localization problem in this paper we treat these directed models using an approximation suggested by morita 1964 j math phys 5 1401 5 in which the proportion of each comonomer is fixed we find that the morita approximation leads to behaviour that is closer to that of the quenched average model and this is particularly interesting in the localization problem where the phase diagram is a tl east qualitatively very similar to that of the quenched average problem we also show that the phase boundaries in the morita approximation are bounds on the locations of the phase boundaries of the quenched model

self averaging in the statistical mechanics of some lattice models we discuss self averaging of thermodynamic properties in some random lattice models in particular we investigate when self averaging in an almost sure sense of the free energy implies self averaging of the energy and heat capacity and we discuss the connection between self averaging in the almost sure sense and self averaging in an l p sense under quite general conditions we show that the average of the finite size heat capacity converges to the second derivative of the limiting quenched average free energy we consider the application of these ideas to the problem of adsorption of a random copolymer at a surface and to some related systems

recursive interpolation algorithm a formalism for solving systems of linear equations i direct methods abstract this paper presents a simple unifying algorithm for solving systems of linear equations solving a system of linear equations will be interpreted as an interpolation problem this new approach leads us to a general algorithm called the recursive interpolation algorithm ria which includes the direct methods and some of the iterative methods a version of the ria with pivoting strategy will be given we will also show how to choose two free sets of parameters in the ria for recovering known direct methods other choices of these parameters yield some new methods

random parking euclidean functionals and rubber elasticity we study subadditive functions of the random parking model previously analyzed by the second author in particular we consider local functions s of subsets of mathbb r d and of point sets that are almost subadditive in their first variable denoting by the random parking measure in mathbb r d and by r the random parking measure in the cube qr r r d we show under some natural assumptions on s that there exists a constant overline s in mathbb r such that lim r to infty frac s q r xi q r lim r to infty frac s q r xi r q r overline s almost surely if zeta mapsto s q r zeta is the counting measure of zeta in qr then we retrieve the result by the second author on the existence of the jamming limit the present work generalizes this result to a wide class of almost subadditive functions in particular classical euclidean optimization problems as well as the discrete model for rubber previously studied by alicandro cicalese and the first author enter this class of functions in the case of rubber elasticity this yields an approximation result for the continuous energy density associated with the discrete model at the thermodynamic limit as well as a generalization to stochastic networks generated on bounded sets

multidimensional random subshifts of finite type and their large deviations i introduce random multidimensional subshifts of finite type which generalize models of spin glasses and establish the almost sure large deviations bounds for gibbs measures there the paper is sequel to ekw where the corresponding results were obtained for deterministic multidimensional subshifts of finite type

diffusion of a heteropolymer in a multi interface medium we consider a heteropolymer consisting of an i i d concatenation of hydrophilic and hydrophobic monomers in the presence of water and oil arranged in alternating layers the heteropolymer is modelled by a directed path left i s i right i in mathbb n 0 where the vertical component lives on mathbb z and the layers are horizontal with equal width the path measure for the vertical component is given by that of simple random walk multiplied by an exponential weight factor that favors matches and disfavors mismatches between the monomers and the medium we study the vertical motion of the heteropolymer as a function of its total length n when the width of the layers is d n and the parameters in the exponential weight factor are such that the heteropolymer tends to stay close to an interface localized regime in the limit as n and under the condition that lim n d n log log n and lim n d n log n 0 we show that the vertical motion is a diffusive hopping between neighboring interfaces on a time scale exp d n 1 o 1 where is computed explicitly in terms of a variational problem an analysis of this variational problem sheds light on the optimal hopping strategy

self avoiding walks on the simple cubic lattice we have substantially extended the series for the number of self avoiding walks and the mean square end to end distance on the simple cubic lattice our analysis of the series gives refined estimates for the critical point and critical exponents our estimates of the exponents and are in good agreement with recent high precision monte carlo estimates and also with recent renormalization group estimates critical amplitude estimates are also given a new improved rigorous upper bound for the connective constant 4 7114 is obtained

exact evaluation of the collapse phase boundary for two dimensional directed polymers the phase boundary of the collapsed phase for a directed polymer on a square lattice in the presence of an attractive wall and monomer monomer interactions is calculated exactly using transfer matrix techniques the position of the multicritical point is also identified

the contraction number of a multigrid method for solving the poisson equation the treatment of a multigrid method in the framework of numerical analysis elucidates that regularity of the solution is not necessary for the convergence of the multigrid algorithm but only for fast convergence for the linear equations which arise from the discretization of the poisson equation a convergence factor 0 5 is established independent of the shape of the domain and of the regularity of the solution

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

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

abs algorithms for linear equations and optimization in this paper we review basic properties and the main achievements obtained by the class of abs methods developed since 1981 to solve linear and nonlinear algebraic equations and optimization problems

a monte carlo study of polymer adsorption random copolymers and random surfaces we consider a model of random copolymer adsorption in which an n edge self avoiding walk in three dimensions interacts with a plane defining a half space to which the walk is confined each vertex of the walk is randomly labelled a with probability p or b with probability 1 p and only vertices labelled a are attracted to the surface plane the system is quenched i e the labelling is fixed and then the thermodynamic properties are computed we use monte carlo methods to investigate the behaviour of this system we observe self averaging of the energy as n increases and investigate the location of the adsorption transition for various values of p in addition we compare the behaviour of this system with that of a homopolymer adsorbing at a randomly heterogeneous surface consisting of two types of sites only one of which interacts with the monomers of the polymer

pulling polymers adsorbed on a striped surface we consider the adsorption of a linear homopolymer on a striped surface where the monomers have different energies of interaction with each of the two kinds of stripe in addition the polymer is subject to a force that tends to desorb the polymer and the force can be applied at different angles we consider both a partially directed walk model and a self avoiding walk model we solve the problem for the partially directed walk problem essentially completely and give phase diagrams in the force temperature plane for various values of the direction in which the force is applied the phase diagrams show re entrance phenomena and for some force directions degenerate ground states we also present a low temperature treatment that helps to interpret the results from the combinatorial solution for the self avoiding walk model we present monte carlo results and show that the phase diagrams for the partially directed walk and self avoiding walk are qualitatively similar and at low temperatures are in quantitative agreement

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

chain polymers near an adsorbing surface we present improved estimates of critical exponents for 3d polymers near a plane adsorbing surface the improvements were possible by simulating rather long chains up to n 2000 by means of a new algorithm apart from minor but significant adjustments in the location of the tricritical point and in the various gamma exponents our main result is that the crossover exponent for critical adsorption is compatible with phi 1 2 this is much lower than previous estimates and suggests that this exponent might be superuniversal

localization of random copolymers and the morita approximation we discuss and analyse the morita approximation for a number of different models of quenched random copolymer localization at the interface between two immiscible liquids we focus on two directed models bilateral dyck paths and bilateral motzkin paths for which this approximation can be carried through analytically we study the form of the phase diagram and find that the morita approximation gives phase boundaries which are qualitatively correct this is also true when a monomer interface interaction is included in the model when this interaction is attractive it can lead to separation of the phase boundaries which is also a feature of the quenched problem we note the existence of non analytic points on the phase boundaries which may reflect tricritical points on the phase boundaries of the full quenched average problem in certain regions of the phase plane this approximation can be extended to the self avoiding walk model

trapping regions for discontinuously coupled systems of evolution variational inequalities and application we consider initial boundary value problems for weakly coupled systems of parabolic equations under coupled nonlinear flux boundary condition both coupling vector fields f q r 2 r 2 and g r 2 r 2 are assumed to be either of competitive or cooperative type but may otherwise be discontinuous with respect to all their arguments the main goal is to provide conditions for the vector fields f and g that allow the identification of regions of existence of solutions so called trapping regions to this end the problem is transformed to a discontinuously coupled system of evolution variational inequalities assuming a generalized outward pointing vector field on the boundary of a rectangle of the dependent variable space the system of evolution variational inequalities is solved via a fixed point problem for some increasing operator in an appropriate ordered banach space the main tools used in the proof are evolution variational inequalities comparison techniques and fixed point results in ordered banach spaces

pruned enriched rosenbluth method simulations of θ polymers of chain length up to 1 000 000 we present an algorithm for simulating flexible chain polymers it combines the rosenbluth rosenbluth method with recursive enrichment although it can be applied also in more general situations it is most efficient for three dimensional theta polymers on the simple cubic lattice there it allows high statistics simulations of chains of length up to n 10 sup 6 for storage reasons this is feasable only for polymers in a finite volume for free theta polymers in infinite volume we present very high statistics runs with n 10000 these simulations fully agree with previous simulations made by hegger and grassberger j chem phys bold 102 6681 1995 with a similar but less efficient algorithm showing that logarithmic corrections to mean field behavior are much stronger than predicted by field theory but the finite volume simulations show that the density inside a collapsed globule scales with the distance from the theta point as predicted by mean field theory in contrast to claims in the work mentioned above in addition to the simple cubic lattice we also studied two versions of the bond fluctuation model but with much shorter chains finally we show that our method can be applied also to off lattice models and illustrate this with simulations of amore model studied in detail by freire ital et al macromolecules bold 19 452 1986 and later work copyright ital 1997 ital the american physical society less

a new taxonomy of conjugate gradient methods abstract in this paper a framework is proposed for conjugate gradient and conjugate direction algorithms based on an indefinite quadratic form by the choice of the hessian g of this form and of another arbitrary matrix k no fewer than nineteen distinct cg algorithms may be succinctly defined four of these are believed to be new the choice of certain other vectors leads either to a two term or a three term recurrence formula and it is shown that to every two term formula there corresponds a three term formula although the converse is not generally true the two matrices g and k also determine the stability characteristics of the two term methods

a numerical approach to copolymers at selective interfaces we consider a model of a random copolymer at a selective interface which undergoes a localization delocalization transition in spite of the several rigorous results available for this model the theoretical characterization of the phase transition has re mained elusive and there is still no agreement about several important issues for example the behavior of the polymer near the phase transition line from a rigorous viewpoint non coinciding upper and lower bounds on the critical line are known in this paper we combine numerical computations with rigorous arguments to get to a better understanding of the phase diagram our main results include various numerical observations that suggest that the critical line lies strictly in between the two bounds a rigorous statistical test based on concentration inequalities and super additivity for determining whether a given point of the phase diagram is in the localized phase this is applied in particular to show that with a very low level of error the lower bound does not coincide with the critical line an analysis of the precise asymptotic behavior of the partition function in the de localized phase with particular attention to the effect of rare atypical stretches in the disorder sequence and on whether or not in the delocalized regime the polymer path has a brownian scaling a new proof of the lower bound on the critical line this proof relies on a characteri zation of the localized regime which is more appealing for interpreting the numerical data

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

constrained conjugate directions methods for design optimization of large systems the constrained steepest descent directions obtained as the solution of a quadratic programming subproblem are used to generate the constrained conjugate directions the resulting algorithm is quite simple and its finite convergence and other properties have been proved for convex quadratic programing problems for general problems instart procedures step size determination and potential constraint strategy are discussed

self averaging sequences in the statistical mechanics of random copolymers a constructive proof is given that certain problems in the statistical mechanics of random copolymers are thermodynamically self averaging the proof relies on the use of normal numbers and potentially gives a method for calculating expectations in the quenched averaged ensemble in particular we give a new proof that adsorbing random copolymers are self averaging

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

stochastic homogenisation of free discontinuity problems in this paper we study the stochastic homogenisation of free discontinuity functionals assuming stationarity for the random volume and surface integrands we prove the existence of a homogenised random free discontinuity functional which is deterministic in the ergodic case moreover by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise subadditive ergodic theorem by akcoglou and krengel we characterise the limit volume and surface integrands in terms of asymptotic cell formulas

self avoiding walks interacting with a surface discusses two models of polymer adsorption in one model a self avoiding walk on a d dimensional lattice interacts with a d 1 dimensional hyperplane in the other model this walk must also lie in or on one side of this hyperplane both models exhibit non analytic behaviour corresponding to a phase transition but these phase transitions do not occur at the same point the authors give numerical estimates of the locations of these transitions

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

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

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

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

a heuristic iterated subspace minimization method with pattern search for unconstrained optimization recently an increasing attention was paid on different procedures for an unconstrained optimization problem when the information of the first derivatives is unavailable or unreliable in this paper we consider a heuristic iterated subspace minimization method with pattern search for solving such unconstrained optimization problems the proposed method is designed to reduce the total number of function evaluations for the implementation of high dimensional problems meanwhile it keeps the advantages of general pattern search algorithm i e the information of the derivatives is not needed at each major iteration of such a method a low dimensional manifold the iterated subspace is constructed and an approximate minimizer of the objective function in this manifold is then determined by a pattern search method numerical results on some classic test examples are given to show the efficiency of the proposed method in comparison with a conventional pattern search method and a derivative free method

quenched invariance principle for simple random walk on percolation clusters we consider the simple random walk on the unique infinite cluster of super critical bond percolation in d with d 2 we prove that for almost every percolation configuration the path distribution of the walk converges weakly to that of non degenerate isotropic brownian motion our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square integrable martingale the size of the deformation expressed by the so called corrector is estimated by means of ergodicity arguments

collapsing and adsorbing polygons the adsorption transition in the phase diagram of a self interacting lattice polygon is examined the polygon has a nearest neighbour contact fugacity and the interaction between the polygon and an impenetrable wall is modelled by a visit fugacity which is conjugate to the number of vertices of the polygon incident with the wall the partition function of this model is z c y z d p v cp c v c y v z c where p c v c is the number of polygons with c nearest neighbour contacts v visits to the wall and n edges and counted up to translations parallel to the wall the limiting free energy of this model is f c y z d limn 1 1 logz c y z and it is known to be a non analytic function of y for each z 1 for all z2 0 1

impedance matrix compression imc using iteratively selected wavelet basis we present a novel approach for the incorporation of wavelets into the solution of frequency domain integral equations arising in scattering problems in this approach we utilize the fact that when the basis functions used are wavelet type functions only a few terms in a series expansion are needed to represent the unknown quantity to determine these dominant expansion functions an iterative procedure is devised the new approach combined with the iterative procedure yields a new algorithm that has many advantages over the presently used methods for incorporating wavelets numerical results which illustrate the approach are presented for three scattering problems

γ convergence and homogenisation for a class of degenerate functionals abstract this paper is on convergence for degenerate integral functionals related to homogenisation problems in the heisenberg group here both the rescaling and the notion of invariance or periodicity are chosen in a way motivated by the geometry of the heisenberg group without using special geometric features these functionals would be neither coercive nor periodic so classic results do not apply all the results apply to the more general case of carnot groups

stochastic reinforcement problems and ergodic theory we study the asymptotic behaviour of dirichlet problems in domains of r 2 bounded by thin layers whose thickness is given by means of an assigned ergodic random function using a capacitary method together with ergodic theorems for additive and superadditive processes we are able to characterise the limit problem precisely

regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity we prove regularity and stochastic homogenization results for certain degenerate elliptic equations in nondivergence form the equation is required to be strictly elliptic but the ellipticity may oscillate on the microscopic scale and is only assumed to have a finite d th moment where d is the dimension in the general stationary ergodic framework we show that the equation homogenizes to a deterministic uniformly elliptic equation and we obtain an explicit estimate of the effective ellipticity which is new even in the uniformly elliptic context showing that such an equation behaves like a uniformly elliptic equation requires a novel reworking of the regularity theory we prove deterministic estimates depending on averaged quantities involving the distribution of the ellipticity which are controlled in the macroscopic limit by the ergodic theorem we show that the moment condition is sharp by giving an explicit example of an equation whose ellipticity has a finite p th moment for every p less than d but for which regularity and homogenization break down while these results are new even for linear equations we prove them in the fully nonlinear context in probabilistic terms our homogenization results correspond to quenched invariance principles for controlled diffusion processes in random media including linear diffusions as well as diffusions controlled by one controller or two competing players

conjugate gradient like algorithms for solving nonsymmetric linear systems this paper presents a unified formulation of a class of the conjugate gradient like algorithms for solving nonsymmetric linear systems the common framework is the petrov galerkin method on krylov subspaces we discuss some practical points concerning the methods and point out some of the interrelations between them 1 introduction in the recent few years a large number of generalizations of the conjugate gradient and conjugate residual methods which are very successful in solving symmetric positive definite linear systems have been proposed for solving nonsymmetric linear systems 3 6 5 7 12 in this paper we present an abstract framework which includes most of these methods and many new ones our goal is to understand the relationships among the methods and to synthesize consider the general linear system

a trapping principle for discontinuous elliptic systems of mixed monotone type abstract we consider discontinuous semilinear elliptic systems with boundary conditions on the individual components of dirichlet neumann type the system is a divergence form generalization of u f u the components of f are required to satisfy monotonicity conditions associated with competitive or cooperative species the latter model defines a system of mixed monotone type we also illustrate the theory via higher order mixed monotone systems which combine competitive and cooperative subunits we seek solutions on special intervals defined by lower and upper solutions associated with outward pointing vector fields it had been shown by heikkila and lakshmikantham that the general discontinuous mixed monotone system does not necessarily admit solutions on an interval defined by lower and upper solutions our result obtained via the tarski fixed point theorem shows that solutions exist for the models described above in the sense of a measurable selection in the principal arguments from a maximal monotone multivalued mapping we use intermediate variational inequalities in the proof applications involving quantum confinement and chemically reacting systems with change of phase are discussed these are natural examples of discontinuous systems

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

a 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

transfer matrix calculation of the exponent γ for two dimensional self avoiding walks we develop two independent transfer matrix methods for the determination of the exponent in the two dimensional self avoiding walk problem our first method is based on the calculation of the correlation length and uses conformal invariance our second method is based on the direct calculation of the moments of the order parameter distribution our results are in good agreement with the conjectured values

doubly coupled systems of parabolic hemivariational inequalities existence and extremal solutions abstract we consider an initial boundary value problem for a quasilinear parabolic system of hemivariational inequalities which is not necessarily coercive the system exhibits full dependence on the gradient of the solution and is doubly coupled on both the source and multivalued terms based on sub supersolutions truncation functions and nonsmooth analysis we establish an existence and enclosure result for weak solutions within a trapping region the existence of extremal solutions is proved without imposing any monotonicity conditions on lower order terms moreover we present a sufficient condition to construct effectively a trapping region and show the existence of positive solutions

free energy and some sample path properties of a random walk with random potential we study the asymptotic behavior of the free energy for a model defined by sinai of one dimensional random walk with random potential in particular we obtain a central limit theorem and a strong law of large numbers for this free energy we use some results on the free energy to study some sample path properties of this random walk which are related respectively to its recurrence and localization some exponents describing the recurrence and localization are found

generalized conjugate directions abstract this paper presents a simple unifying framework for a wide class of conjugate directions algorithms whose iterates minimize some quadratic functional over a subspace our approach is motivated by its advantages for nonlinear minimization but the purpose of this paper is to present the greatly simplified convergence analysis that results for the linear case

a parallel algorithm for the enumeration of self avoiding polygons on the square lattice we have developed a parallel algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 110 we have also extended the series for the first 10 area weighted moments and the radius of gyration to 100 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 530 31 3 biased and the critical exponent 0 500 000 1 2 unbiased in addition we obtain very accurate estimates for the leading amplitudes confirming to a high degree of accuracy various predictions for universal amplitude combinations

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

configurational statistics of a disordered polymer chain a problem of the configurational properties of a long flexible polymer chain with a quenched disorder is considered the chain is assumed to be randomly constructed from monomers of two different kinds with different constants for the two body interaction near the theta point i e when the average interaction of monomers is small the spatial correlation of the repulsive and attractive monomers of different kinds leads to an increase of effects of the disorder on large scales there is also the competing effect of the repulsive three body interaction which tends to screen the effects of disorder on large scales for both effects the upper critical dimension is dc 3 a solution of the renormalisation group equation indicates that there always exists a critical scale at which the relative dispersion of sizes of polymers with different random sequences of monomers becomes of the order of unity the magnitude of this critical scale depends strongly on the relation between the constant which characterises the dispersion of the two body interaction b0 and the constant of the repulsive three body interaction v0 if b0 3 32v0 at each physically attainable scale the effects of screening are prevalent and the dispersion of sizes of polymers with different sequences of monomers is small near the theta point if the reverse inequality holds the dispersion of sizes becomes of the order of unity near the theta point

a trapping principle and convergence result for finite element approximate solutions of steady reaction diffusion systems we consider nonlinear elliptic systems with mixed boundary conditions on a convex polyhedral domain rn these are nonlinear divergence form generalizations of u f u where f is outward pointing on the trapping region boundary the motivation is that of applications to steady state reaction diffusion systems also included are reaction diffusion convection systems which satisfy the einstein relations for which the cole hopf transformation is possible for maximum generality the theory is not tied to any specific application we are able to demonstrate a trapping principle for the piecewise linear galerkin approximation defined via a lumped integration hypothesis on integrals involving f by use of variational inequalities results of this type have previously been obtained for parabolic systems by estep larson and williams and for nonlinear elliptic equations by karatson and korotov recent minimum and maximum principles have been obtained by jungel and unterreiter for nonlinear elliptic equations we make use of special properties of the element stiffness matrices induced by a geometric constraint upon the simplicial decomposition this constraint is known as the non obtuseness condition it states that the inward normals associated with an arbitrary pair of an element s faces determine an angle with nonpositive cosine dr g nescu dupont and scott have constructed an example for which the discrete maximum principle fails if this condition is omitted we also assume vertex communication in each element in the form of an irreducibility hypothesis on the off diagonal elements of the stiffness matrix there is a companion convergence result which yields an existence theorem for the solution this entails a consistency hypothesis for interpolation on the boundary and depends on the tabata construction of simple function approximation based on barycentric regions

contraction numbers for additive correction methods abstract generic contraction numbers are derived for a class of additive correction methods based on orthogonal projection the only assumption on the range of the projector is that it contains the residual this generalizes previous work of the authors on positive definite operators and provides convergence results for restarted generalized conjugate gradient methods under a variety of conditions

collapse transition and crossover scaling for self avoiding walks on the diamond lattice a monte carlo study of self avoiding walks on the diamond lattice is presented this model incorporates some of the steric effects and short range stiffness of real alkanes and for a nearest neighbour attractive interaction epsilon is found to have a collapse transition at kb theta epsilon 2 25 or 0 05 the behaviour of the chains in the vicinity of this theta temperature is analysed with the help of recent crossover scaling theories it is shown that for finite chain length n there is a rather broad theta region where the chains behaviour is quasi ideal the width of this region wtheta behaves as omega 0 varies as n 12 consistent with the blob picture the peak of the specific heat occurs at the boundary between the theta region and the region of collapsed chains the authors also give rough estimates for the scaling functions describing the crossovers of the end to end distance and structure factor of the chains

self avoiding walks in wedges the authors consider the number of self avoiding walks confined to a subset zd f of the d dimensional hypercubic lattice zd such that the coordinates x1 x2 xd of each vertex in the walk satisfy x1 or 0 and 0 or xk or fk x1 for k 2 3 d they show that if fk x to infinity as x to infinity the connective constant of walks in zd f is identical to the convective constant of walks in zd they also explore conditions on fk which lead to a smaller connective constant for walks in zd f and in particular consider walks between two parallel d 1 dimensional hyperplanes finally they contrast some of these results with recent work by grimmett on percolation on subsets of the square lattice

copolymers at selective interfaces new bounds on the phase diagram we investigate the phase diagram of disordered copolymers at the interface between two selective solvents and in particular its weak coupling behavior encoded in the slope mc of the critical line at the origin we focus on the directed walk case which has turned out to be in spite of the apparent simplicity extremely challenging in mathematical terms the partition function of such a model does not depend on all the details of the markov chain that models the polymer but only on the time elapsed between successive returns to zero and on whether the walk is in the upper or lower half plane between such returns this observation leads to a natural generalization of the model in terms of arbitrary laws of return times the most interesting case being the one of return times with power law tails with exponent 1 1 2 in the case of the symmetric random walk the main results we present here are

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

localization of a random copolymer at an interface an exact enumeration study we consider a self avoiding walk on the simple cubic lattice as a model of localization of a random copolymer at an interface between two immiscible liquids the vertices of the walk are coloured a or b randomly and independently the two liquid phases are represented by the two half spaces z 0 and z 0 and the plane z 0 corresponds to the interface between the two liquids the energy depends on the numbers of a vertices with positive z coordinate and b vertices with negative z coordinate in addition there is a vertex interface interaction irrespective of the colour of the vertex we use exact enumeration and series analysis techniques to investigate the form of the phase diagram and how it changes as the magnitude of the vertex interface interaction changes

trapping region for discontinuous quasilinear elliptic systems of mixed monotone type we consider discontinuous quasilinear elliptic systems with nonlinear boundary conditions of mixed dirichlet robin type on the individual components the system considered is of the general form au f u h where a is a quasilinear elliptic operator of leray lions type u u 1 u2 and the vector fleld f f 1 f2 i s assumed to be of mixed monotone type associated with competitive or cooperative species the vector fleld f may be discontinuous with respect to all its arguments the main goal is to prove the existence of solutions within the so called trapping region furthermore if in addition the components fk are continuous in their ofidiagonal nonprincipal arguments one can show the compactness of the solution set within the trapping region the main tools used in the proof of our main result are variational inequalities truncation and comparison techniques employing special test functions and tarski s flxed point theorem on complete lattices two applications of the theory developed in this paper are provided the flrst application deals with the steady state transport of two species of opposite charge within a physical channel and in the second application a uid medium is considered which may undergo a change of phase and which acts as a carrier for certain solute species

a monte carlo study of adsorption of random copolymers on random surfaces we study the adsorption problem of a random copolymer on a random surface in which a self avoiding walk in three dimensions interacts with a plane defining a half space to which the walk is confined each vertex of the walk is randomly labelled a with probability pp or b with probability 1 pp and only vertices labelled a are attracted to the surface plane each lattice site on the plane is also labelled either a with probability ps or b with probability 1 ps and only lattice sites labelled a interact with the walk we study two variations of this model in the first case the a vertices of the walk interact only with the a sites on the surface in the second case the constraint of selective binding is removed that is any contact between the walk and the surface that involves an a labelling either from the surface or from the walk is counted as a visit to the surface the system is quenched in both cases i e the labellings of the walk and of the surface are fixed as thermodynamic properties are computed we present monte carlo simulation results which provide evidence for second order transitions in both cases we observe that in both cases the adsorption location depends on both ps and pp the dilution of the interactive sites on the surface and the walk we compare critical properties for the two cases and find that the adsorption location varies for different models while the crossover exponent is independent of the details of the models

low temperature properties of directed walks with random self interactions we consider a simple model for a chain with random self interactions each monomer on the chain carries a charge which is a random variable and the configurations of the chain are directed walks we show that at zero temperature the end to end distance r and the energy e of chains with total charge q approximately nx scale as r approximately nnu x and e approximately ntheta x the exponents thus depend on the total charge through the parameter x these exponents are measured numerically and compared to the predictions of some simple arguments at a finite temperature r is extensive and appears to be independent of x

a note on the restarted cg method and reduced space additive correction abstract this paper develops a general theory of the reduced space additive correction method rsac which can also be viewed as a two grid method in particular we show that the restarted conjugate gradient method is a two grid method consisted of a richardson presmoothing phase followed by a projection or error correction step in a pseudo residual space

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

phenomenological renormalisation of the self avoiding walk in two dimensions phenomenological renormalisation is used to calculate the exponent v and the connective constant of the self avoiding walk problem on a square lattice a transfer matrix technique is developed for the polymer problem the results indicate that flory s value v 0 75 is true in two dimensions to extremely high accuracy

solutions to variational inequalities of parabolic type abstract the existence of strong solutions to a kind of variational inequality of parabolic type is investigated by the theory of semigroups of linear operators as an application an abstract semi permeable media problem is studied

higher order morita approximations for random copolymer adsorption random linear copolymers are linear 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 even for simple configurational models the quenched model is too difficult to solve analytically the morita approximation is a partial annealing procedure which yields upper bounds on the quenched average free energy in this paper we consider higher order morita approximations in which we control correlations to various orders between neighbouring monomers along the polymer chain we consider different approaches for incorporating correlations and apply these to motzkin and dyck path models of the adsorption of a random copolymer at a surface we also present lower bounds which along with the morita bounds determine the limiting quenched average free energy for adsorption very precisely at low temperatures

large sparse continuation problems we study and develop efficient and versatile predictor corrector continuation methods for large sparse problems the first object is to show how special solving methods for a large sparse linear systems can be incorporated into the basic steps of a continuation method next we describe how to use a special nonlinear conjugate gradient method to perform the corrector phase it is shown how such methods can be used to detect bifurcation points and how to trace bifurcating solution branches by using local perturbations finally a numerical example involving bifurcating branches of a nonlinear eigenvalue problem is given

convergence of conjugate residual like methods to solve linear equations convergence results are obtained for a class of conjugate residual like algorithms to solve a real linear system of algebraic equations the symmetric part of the matrix encountered is assumed to be positive definite some available results are shown to be special cases

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

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

a taxonomy for conjugate gradient methods the conjugate gradient method of hestenes and stiefel is an effective method for solving large sparse hermitian positive definite hpd systems of linear equations ax b generalizations to non hpd matrices have long been sought the recent theory of faber and manteuffel gives necessary and sufficient conditions for the existence of a cg method this paper uses these conditions to develop and organize such methods it is shown that any cg method for ax b is characterized by an hpd inner product matrix b and a left preconditioning matrix c at each step the method minimizes the b norm of the error over a krylov subspace this characterization is then used to classify known and new methods finally it is shown how eigenvalue estimates may be obtained from the iteration parameters generalizing the well known connection between cg and lanczos such estimates allow implementation of a stopping criterion based more nearly on the true error

conjugate gradient methods for continuation problems ii we study linear conjugate gradient cg methods for large sparse continuation problems first we show how the cg methods can be incorporated as linear solvers in the context of a continuation method next we indicate how the preconditioned generalized minimal residual algorithm may be used to test for simple and multiple bifurcations without solving the bordered linear systems sample numerical results are reported some concluding remarks concerning the performance of three term cg methods are given

localization transition for a polymer near an interface consider the directed process i s i where the second component is simple random walk on mathbb z s 0 0 define a transformed path measure by weighting each n step path with a factor exp lambda sum 1 leq i leq n omega i h sign s i here omega i i geq 1 is an i i d sequence of random variables taking values pm 1 with probability 1 2 acting as a random medium while lambda in 0 infty and h in 0 1 are parameters the weight factor has a tendency to pull the path towards the horizontal because it favors the combinations s i 0 omega i 1 and s i 0 omega i 1 the transformed path measure describes a heteropolymer consisting of hydrophylic and hydrophobic monomers near an oil water interface we study the free energy of this model as n to infty and show that there is a critical curve lambda to h c lambda where a phase transition occurs between localized and delocalized behavior in the vertical direction we derive several properties of this curve in particular its behavior for lambda downarrow 0 to obtain this behavior we prove that as lambda h downarrow 0 the free energy scales to its brownian motion analogue

experiments with conjugate gradient algorithms for homotopy curvetracking there are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points i e with probability one the essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map hompack is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve and has separate routines for dense and sparse jacobian matrices the hompack algorithms for sparse jacobian matrices use a preconditioned conjugate gradient algorithm for the computation of the kernal of the homotopy jacobian matrix a required linear algebra step for homotopy curve tracking here variants of the conjugate gradient algorithm are implemented in the context of homotopy curve tracking and compared with craig s preconditioned conjugate gradient method used in hompack the test problems used include actual large scale sparse structural mechanics problems

a model of continuous polymers with random charges we study a model of polymers with random charges the possible shapes of the polymer are represented by the sample paths of a brownian motion and the cumulative charge distribution along a polymer is modeled by a realization of a brownian bridge charges interact through a general positive definite two body potential we study the infinite volume free energy density for a fixed realization of the brownian motion this is not self averaging but shows on the contrary a sample dependence through the local time of the brownian motion we obtain an explicit series representation for the free energy density this has a finite radius of convergence but the free energy is nevertheless analytic in the inverse temperature in the physical domain

on the density of state for the quantum percolation problem for the site dilution model on the hypercubic lattice zd d or 2 the authors examine the density of states for the tight binding hamiltonian projected onto the infinite cluster it is shown that with probability one the corresponding integrated density of states is discontinuous on a set of energies which is dense in the band this result is proved by constructing states supported on finite regions of the infinite cluster analogous to the kirkpatrick and eggarter zero energy molecular state