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

three friendly walkers more than 15 years ago guttmann and voge 2002 j stat plan inference 101 107 introduced a model of friendly walkers since then it has remained unsolved in this paper we provide the exact solution to a closely allied model which essentially only differs in the boundary conditions the exact solution is expressed in terms of the reciprocal of the generating function for vicious walkers which is a d finite function however ratios of d finite functions are inherently not d finite and in this case we prove that the friendly walkers generating function is the solution to a non linear differential equation with polynomial coefficients it is in other words d algebraic we find using numerically exact calculations a conjectured expression for the generating function of the original model as a ratio of a d finite function and the generating function for vicious walkers we obtain an expression for this d finite function in terms of a hypergeometric function with a rational pullback and its first and second derivatives

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

carl s inequality for quasi banach spaces we prove that for any two quasi banach spaces x and y and any 0 there exists a constant 0 such that sup1 k n k ek t sup1 k n k ck t holds for all linear and bounded operators t x y here ek t is the k th entropy number of t and ck t is the k th gelfand number of t for banach spaces x and y this inequality is widely used and well known as carl s inequality for general quasi banach spaces it is a new result

nonperturbative tricritical exponents of trails i exact enumeration on a triangular lattice as the fugacity of intersections in trails intersecting but non overlapping lattice walks is increased the configurations change from swollen ones with the scaling exponents of self avoiding walks to compact ones separating the two regimes is a potentially new tricritical point with no perturbative renormalisation fixed point associated with it supporting evidence for the existence of a tricritical point its likely location and exponents are computed for the first time from exact enumeration of all trails up to length of fifteen lattice constants on the triangular lattice the divergence of the specific heat indicates the location of the tricritical point generalised ratio and pade methods are used to extract the scaling exponents for the number of configurations and their end to end distance

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

global attractor for a nonlinear plate equation with supported boundary conditions abstract in this paper we consider a two dimensional nonlinear equation w t t d 2 w e w t n 1 t 2 w x 2 d x d y w x x n 2 t 2 w y 2 d x d y w y y 0 which arises from the model of the viscoelastic thin rectangular plate with four edges supported by virtue of galerkin method combined with the priori estimates we prove the existence and uniqueness of the global solution under initial boundary data for the above equation especially the existence of the bounded absorbing set in space e and the existence of the global attractor of system is also obtained

interpolation of compact operators by general interpolation methods abstract let a linear operator t act compactly from a banach couple x x 0 x 1 in a banach lattice couple b b 0 b 1 if both b i have absolutely continuous norm then any interpolation functor f preserves two sided compactness of t the same holds if only one of b i satisfies the above condition and f is a quasipower regular functor the similar results are true for operators acting from b in x under the above formulated assumptions for simultaneously b and its dual b

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 variational formula for the free energy of the partially directed polymer collapse long linear polymers in dilute solutions are known to undergo a collapse transition from a random coil expand itself to a compact ball fold itself up when the temperature is lowered or the solvent quality deteriorates a natural model for this phenomenon is a 1 1 dimensional self interacting and partially directed self avoiding walk in this paper we develop a new method to study the partition function of this model from which we derive a variational formula for the free energy this variational formula allows us to prove the existence of the collapse transition and to identify the critical temperature in a simple way we also provide a probabilistic proof of the fact that the collapse transition is of second order with critical exponent 3 2

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

specific heat and crossover exponents at the tricritical collapse of trails from the exact enumeration on a triangular lattice of trail configurations at the tricritical regime as the fugacity for intersections is increased the authors find phi t 0 68 8 for the crossover exponent and hence alpha t 0 5 2 for the specific heat exponent it indicates the collapse transition of trails to differ from that of polymer chains at the theta point in two dimensions

minimal and maximal methods of interpolation abstract we show that several interpolation functors including the widely used real and complex methods are minimal or maximal extensions from a single couple of banach spaces various consequences are drawn from this property

entropy numbers of embeddings of schatten classes abstract let 0 p q and denote by s p n and s q n the corresponding finite dimensional schatten classes we prove optimal bounds up to constants only depending on p and q for the entropy numbers of natural embeddings between s p n and s q n this complements the known results in the classical setting of natural embeddings between finite dimensional l p spaces due to schutt edmunds triebel triebel and guedon litvak kuhn we present a rather short proof that uses all the known techniques as well as a constructive proof of the upper bound in the range n n n 2 that allows deeper structural insight and is therefore interesting in its own right our main result can also be used to provide an alternative proof of recent lower bounds in the area of low rank matrix recovery

the diffusive phase of a model of self interacting walks we consider simple random walk onzd perturbed by a factor exp t pjt wheret is the length of the walk and j t sum nolimits 0 leqslant i 0 ford 2 the diffusion constant is equal to 1 but ford 2 it is renormalized ford 1 andp 3 2 we prove diffusion for all real positive or negative ford 2 the scaling limit is brownian motion but ford 2 it is the edwards model with the wrong sign of the coupling when 0 which governs the limiting behaviour the latter arises since for p frac 4 d 2 t p j t is the discrete self intersection local time this establishes existence of a diffusive phase for this model existence of a collapsed phase for a very closely related model has been proven in work of bolthausen and schmock

one sided compactness results for aronszajn gagliardo functors interpolating compactness properties of operators is a long standing and important problem in this paper the authors consider the problem in a very general setting of aronszajn gagliardo functors in simplest terms they show that if t a0 b0 is compact and t a1 b1 is bounded then t is compact on some interpolation spaces constructed in the aronszajn gagliardo methods these methods do not include the complex interpolation method but the authors show that if the two couples are formed by banach lattices then the theorem holds in the complex method as well additional results are in the context of interpolation of compactness properties in the context of n tuples of banach spaces the paper includes a number of related interesting results

entropy numbers and interpolation this paper settles a long standing question by showing that in certain circumstances the entropy numbers of a map do not behave well under real interpolation to do this lemmas of combinatorial type are established and used to obtain lower bounds for the entropy numbers of a particular diagonal map acting between lorentz sequence spaces these lower bounds contradict the estimates from above that would be obtained if the behaviour of entropy numbers under real interpolation was as good as conjectured the paper also provides sharp two sided estimates of the entropy number en t of diagonal operators t l p rightarrow l q t left a k right k 1 infty lambda k a k k 1 infty where 0 p q and lambda i i 1 infty is a non increasing sequence of non negative numbers with i n for all i n

study of the θ point by enumeration of self avoiding walks on the triangular lattice the author reports series expansion analyses of self avoiding walks with nearest neighbour bond interactions the estimates 2 nu t 1 07 or 0 05 and phi 0 64 or 0 05 for the correlation and crossover exponents at the theta point were obtained by examining the number of walks and the end to end distance data up to 16 steps on the triangular lattice

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

lattice trails i exact results considers the problem of lattice trails introduced by malakis 1975 1976 the author proves the existence of a finite connective constant and establishes a result for the growth of n step trails tn analogous to the best known result for self avoiding walks tn lambda n expt o n for the honeycomb d 2 and lave s lattice d 3 the author establishes a counting theorem from which one can deduce the exact value of the connective constant lambda for the honeycomb lattice lambda 2 2 square root 2 further it follows that the trail problem is in the same universality class as the self avoiding walk problem for those lattices an exact amplitude relation between trails and self avoiding walks and between dumb bells and trails and self avoiding walks is established the non existence of a counting theorem for arbitrary lattices is established an inequality for the triangular lattice connective constant is proved a high density expansion for lambda for the d dimensional hybercubic lattice is also obtained the author argues that contrary to recent suggestions the model is in the same universality class as the self avoiding walk model

twist in an exactly solvable directed lattice ribbon we investigate the transition between a twisted regime and a disordered regime in a directed ribbon model on a cubic lattice a fugacity corresponding to an interaction which models half twists in the ribbon is introduced and the interacting model is solved exactly our results suggest that conformational entropy and a local interaction which induces twist are key ingredients to model qualitatively the crossover behavior between a twisted helical regime and a denatured regime in duplex biopolymers such as dna

limiting variants of krasnoselʼskiĭʼs compact interpolation theorem abstract we illustrate how limiting variants of krasnosel ski s compact interpolation theorem may be obtained as a special case of our results it follows that compactness is preserved on certain lorentz zygmund spaces close to the end point lebesgue spaces

on compactness of the difference of composition operators abstract let and be analytic self maps of the unit disc and denote by c and c the induced composition operators the compactness and weak compactness of the difference t c c are studied on h p spaces of the unit disc and l p spaces of the unit circle it is shown that the compactness of t on h p is independent of p 1 the compactness of t on l 1 and m the space of complex measures is characterized and examples of and are constructed such that t is compact on h 1 but non compact on l 1 other given results deal with l weakly compact counterparts of the previous results and a conjecture of j e shapiro

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

real interpolation and compact linear operators we prove that if t a0 b0 is compact and t a1 b1 is compact or t a1 b1 is bounded and then given any and q with 0 1 and 0 q it follows that t a0 a1 0 q b0 b1 0 q is also compact here a0 a1 0 q and b0 b1 0 q are the usual real interpolation spaces

on the analyticity properties of scaling functions in models of polymer collapse we consider the mathematical properties of the generating and partition functions in the two variable scaling region about the tricritical point in some models of polymer collapse we concentrate on models that have a similar behaviour to that of interacting partially directed self avoiding walks ipdsaw in two dimensions however we do not restrict the discussion to that model after describing the properties for a general class of models and stating exactly what we mean by scaling we prove the following theorem if the generating function of finite size partition functions has a tricritical cross over scaling form around the theta point and the associated tricritical scaling function g has a finite radius of convergence then the partition function has a finite size scaling form and importantly the finite size scaling function f is an entire function in the ipdsaw case we have an explicit representation of the finite size scaling function we point out that given our description of tricritical scaling this theorem should apply mutatis mutandis to a wider class of theta point models we discuss the result in relation to the edwards model of polymer collapse for which it has recently been argued that the finite size scaling functions are not entire

collapse of two dimensional linear polymers we study the collapse transition of a two dimensional very long polymer the model we consider is a lattice model where the chain is represented by a self avoiding walk with nearest neighbor attraction by using the transfer matrix technique we calculate exactly the thermal and geometrical properties of the polymer on strips of finite width we then use finite size scaling to determine the values of the tricritical point exponentsv1 0 55 0 01vu 1 15 0 15 1 v1 1 80 0 05 we compare these results to the other values already proposed in the literature

interpolation of fredholm operators we prove novel results on interpolation of fredholm operators including an abstract factorization theorem the main result of this paper provides sufficient conditions on the parameters 0 1 and q 1 under which an operator a is a fredholm operator from the real interpolation space x0 x1 q to y0 y1 q for a given operator a x0 x1 y0 y1 between compatible pairs of banach spaces such that its restrictions to the endpoint spaces are fredholm operators these conditions are expressed in terms of the corresponding indices generated by the k functional of elements from the kernel of the operator a in the interpolation sum x0 x1 if in addition a is invertible operator on endpoint spaces then these conditions are also necessary we apply these results to present a solution of the variant of a long standing problem of lions magenes for the real interpolation method we also discuss some applications to the spectral theory of operators as well as to perturbation of the hardy operator by identity on weighted lp spaces

examination of the theta point from exact enumeration of self avoiding walks self avoiding walks on the square lattice with nearest neighbour attractive interactions are investigated as a model for the two dimensional version of a polymer in dilute solution temperature dependences of the exponent nu and the free energy of the chain are estimated from the exact enumeration data for up to 20 steps the value of nu at the theta point disagrees with the mean field theory the end distance distribution function at the theta point is also examined

on an extreme class of real interpolation spaces we investigate the limit class of interpolation spaces that comes up by the choice 0 in the definition of the real method these spaces arise naturally interpolating by the j method associated to the unit square their duals coincide with the other extreme spaces obtained by the choice 1 we also study the behavior of compact operators under these two extreme interpolation methods moreover we establish some interpolation formulae for function spaces and for spaces of operators

stability and exponential convergence for the boltzmann equation we prove existence uniqueness and stability for solutions of the nonlinear boltzmann equation in a periodic box in the case when the initial data are sufficiently close to a spatially homogeneous function the results are given for a range of spaces including l1 and extend previous results in l for the non homogeneous equation as well as the more developed lp theory for the spatially homogeneous boltzmann equation

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

critical exponents from nonlinear functional equations for partially directed cluster models we present a method for the derivation of the generating function and computation of critical exponents for several cluster models staircase bar graph and directed column convex polygons as well as partially directed self avoiding walks starting with nonlinear functional equations for the generating function by linearizing these equations we first give a derivation of the generating functions the nonlinear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz alternatively taking the continuum limit leads to nonlinear differential equations from which one can extract the scaling function we find that all the above models are in the same universality class with exponents y 1 2 7 1 3 and b 2 3 all models have as their scaling function the logarithmic derivative of the airy function

elasticity of two dimensional filaments with constant spontaneous curvature we study the mechanical property of a two dimensional filament with constant spontaneous curvature and under uniaxial applied force we derive the equation that governs the stable shape of the filament and obtain analytical solutions for the equation we find that for a long filament with positive initial azimuth angle the azimuth angle is the angle between x axis and the tangent of the filament and under large stretching force the azimuth angle is a two valued function of the arclength decreases first and then increases with increasing arclength otherwise the azimuth angle is a monotonic function of arclength at finite temperature we derive the differential equation that governs the partition function and find exact solution of the partition function for a filament free of force we obtain closed form expressions on the force extension relation for a filament under low force and for a long filament under strong stretching force our results show that for a biopolymer with moderate length and not too small spontaneous curvature the effect of the spontaneous curvature cannot be replaced by a simple renormalization of the persistence length in the wormlike chain model

directed paths in a wedge models of directed paths have been used extensively in the scientific literature to model linear polymers in this paper we examine a directed path model of a linear polymer in a confining geometry a wedge we are particularly interested in cn the number of directed lattice paths of length n steps which take steps in the north east and south east directions and are confined in the wedge y x p where p is an integer we examine the case p 2 in detail and show that the generating function satisfies a functional equation with quartic kernel we give a formal solution by using the kernel method show that this model is not d finite and determine asymptotics to leading order for cn in particular the number of paths of length n in the wedge y x 2 is given by where the constant 0 67874 can be determined to arbitrary accuracy with little computational effort

some reiteration results for interpolation methods defined by means of polygons we continue the research on reiteration results between interpolation methods associated to polygons and the real method applications are given to n tuples of function spaces or spaces or hounded linear operators and banach algebras

finite length scaling of collapsing directed walks we study the finite length scaling of self interacting partially directed self avoiding walks utilising enumeration data up to a total length of 6000 steps this facilitates the evaluation of the numerical techniques available for calculating exponents at the point and in the collapsed phase of walk type models another consequence is the conjecture of a new scaling theory for the collapsed region of the phase diagram and the suggestion that this should be applicable to the wider range of undirected problems including interacting saw we provide a phenomenological picture of the phase transition in terms of the condensation of droplets that allows us to understand the various length scales involved in the problem pacs numbers 05 50 q 05 70 fh 61 41 e email prel aleks brak tonyg mundoe maths mu oz au

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

stochastic nonlinear beam equations an extensible beam equation with a stochastic force of a white noise type is studied lyapunov functions techniques being used to prove existence of global mild solutions and asymptotic stability of the zero solution

exact solution of the discrete 1 1 dimensional sos model with field and surface interactions we present the solution of a linear solid on solid sos model configurations are partially directed walks on a two dimensional square lattice and we include a linear surface tension a magnetic field and surface interaction terms in the hamiltonian there is a wetting transition at zero field and as expected the behavior is similar to a continuous model solved previously the solution is in terms ofq series most closely related to theq hypergeometric functions1 1

on compactness results of lions peetre type for bilinear operators abstract let a a 0 a 1 b b 0 b 1 be banach couples let e be a banach space and let t be a bilinear operator such that t a b e m j a a j b b j for a a 0 a 1 b b 0 b 1 j 0 1 if t a j b j e compactly for j 0 or 1 we show that t may be uniquely extended to a compact bilinear operator from the complex interpolation spaces generated by a and b to e furthermore the corresponding result for the real method is given and we also study the case when e is replaced by a couple e 0 e 1 of banach function spaces on the same measure space

disjointly strictly singular operators and interpolation interpolation properties of the class of disjointly strictly singular operators on banach lattices are studied we also give some applications to compare the lattice structure of two rearrangement invariant function spaces in particular we obtain suitable analytic characterisations of when the inclusion map between two orlicz function spaces is disjointly strictly singular

the power of convex relaxation near optimal matrix completion this paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries this is known as the matrix completion problem and comes up in a great number of applications including the famous netflix prize and other similar questions in collaborative filtering in general accurate recovery of a matrix from a small number of entries is impossible but the knowledge that the unknown matrix has low rank radically changes this premise making the search for solutions meaningful this paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever the information theoretic limit more importantly the paper shows that under certain incoherence assumptions on the singular vectors of the matrix recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit up to logarithmic factors this convex program simply finds among all matrices consistent with the observed entries that with minimum nuclear norm as an example we show that on the order of nr log n samples are needed to recover a random n x n matrix of rank r by any method and to be sure nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog n

statistical mechanics of directed models of polymers in the square lattice directed square lattice models of polymers and vesicles have received considerable attention in the recent mathematical and physical sciences literature these are idealized geometric directed lattice models introduced to study phase behaviour in polymers and include dyck paths partially directed paths directed trees and directed vesicles models directed models are closely related to models studied in the combinatorics literature and are often exactly solvable they are also simplified versions of a number of statistical mechanics models including the self avoiding walk lattice animals and lattice vesicles the exchange of approaches and ideas between statistical mechanics and combinatorics have considerably advanced the description and understanding of directed lattice models and this will be explored in this review the combinatorial nature of directed lattice path models makes a study using generating function approaches most natural in contrast the statistical mechanics approach would introduce partition functions and free energies and then investigate these using the general framework of critical phenomena generating function and statistical mechanics approaches are closely related for example questions regarding the limiting free energy may be approached by considering the radius of convergence of a generating function and the scaling properties of thermodynamic quantities are related to the asymptotic properties of the generating function in this review the methods for obtaining generating functions and determining free energies in directed lattice path models of linear polymers is presented these methods include decomposition methods leading to functional recursions as well as the temperley method that is implemented by creating a combinatorial object one slice at a time a constant term formulation of the generating function will also be reviewed the thermodynamic features and critical behaviour in models of directed paths may be informative about the underlying properties that determine phase diagrams for wider classes of models including physical models of polymers of particular interest are adsorption and collapse transitions in models of polymers and copolymers the properties of thermodynamic quantities in those models are described by tricritical scaling this is reviewed for directed path models and the generating function approaches can be used to apply tricritical scaling to models of adsorbing inflating and collapsing directed lattice paths critical exponents for a variety of models can be obtained in this manner and with it a better understanding and a classification of the models

stabilizability of coupled wave equations in parallel under various boundary conditions stability of the wave equations coupled in parallel suspension system is investigated for free and fixed boundary conditions by energy dissipation boundary controller s to this end the exponential stability criterion and rouche s theorems are used the former is utilized for uniformly exponentially decaying systems and the latter otherwise the results are characterized by the rate of convergence of the solutions and the system parameters for the stated boundary conditions

reiteration formulae for interpolation methods associated to polygons abstract we study spaces generated by applying the interpolation methods defined by a polygon to an n tuple of real interpolation spaces with respect to a fixed banach couple x y we show that if the interior point of the polygon does not lie in any diagonal of then the interpolation spaces coincide with sums and intersections of real interpolation spaces generated by x y applications are given to n tuples formed by lorentz function spaces and besov spaces moreover we show that results fail in general if is in a diagonal

description of logarithmic interpolation spaces by means of the j functional and applications we work with logarithmic interpolation methods a0 a1 q a where 0 or 1 on the contrary to the case 0 1 we show that their description in terms of the j functional changes depending on the relationship between q and a and that there is no description in a certain range then we use these j descriptions to investigate the behavior of compact operators and weakly compact operators under logarithmic interpolation methods in particular we extend a recent compactness result of edmunds and opic for operators between lp spaces over finite measure spaces to finite measure spaces we also determine the dual of a0 a1 q a when 0 or 1

scaling in the collapsed polymer phase exact results the author derives an exact scaling form for the low temperature partition function in a model of polymer collapse this confirms series work and so gives the exponents sigma 1 2 gamma 1 4 and chi 3 4 exactly the model considered is a variant of the self interacting partially directed self avoiding walk in two dimensions

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

adsorption and collapse of self avoiding walks and polygons in three dimensions we consider self avoiding walks and polygons on the simple cubic lattice confined to the half space and interacting with the plane z 0 in addition there is a short range vertex vertex interaction in the walk or polygon which can lead to a collapse transition we explore the interaction between collapse and adsorption in these systems and discuss the form of the phase diagram key results include a proof of the existence of an adsorption transition for polygons for every value of the vertex vertex interaction a corresponding proof for walks when the vertex vertex interaction term is repulsive and a proof that if polygons exhibit a collapse transition then the phase boundary between the expanded and desorbed phase and the collapsed and desorbed phase must be a straight line

interpolation of compact operators the multidimensional case we investigate how compact operators behave under j and k interpolation methods for n spaces and two parameters first we study those methods relationship with those already existing in the literature estimates for the norms of interpolated operators examples characterization as aronszajn gagliardo functors we also describe the relationship between sparr and fernandez methods and we derive sharp estimates for the norms of interpolated operators in fernandez case then we investigate the behaviour of compact operators we begin with the case when one of the n tuples reduces to a single banach space and later we treat the general case by means of the approach developed in 8

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

two dimensional self avoiding walk with hydrogen like bonding phase diagram and critical behaviour the phase diagram for a two dimensional self avoiding walk model on the square lattice incorporating attractive short ranged interactions between parallel sections of walk is derived using numerical transfer matrix techniques the model displays a collapse transition in contrast to the standard point model the transition is first order the phase diagram in the full fugacity temperature plane displays an additional transition line when compared to the point model as well as a critical transition at finite temperature in the hamiltonian walk limit

neighborhoods on the grassmannian of marginals with bounded isotropic constant abstract we show that for any isotropic log concave probability measure on r n for every e 0 every 1 k n and any e g n k there exists f g n k with d e f e and l f c e

square lattice self avoiding walks and biased differential approximants the model of self avoiding lattice walks and the asymptotic analysis of power series have been two of the major research themes of tony guttmann in this paper we bring the two together and perform a new analysis of the generating functions for the number of square lattice self avoiding walks and some of their metric properties such as the mean square end to end distance the critical point x c for self avoiding walks is known to a high degree of accuracy and we utilise this knowledge to undertake a new numerical analysis of the series using biased differential approximants the new method is major advance in asymptotic power series analysis in that it allows us to bias differential approximants to have a singularity of order q at x c when biasing at x c with the analysis yields a very accurate estimate for the critical exponent thus confirming the conjectured exact value to eight significant digits and removing a long standing minor discrepancy between exact and numerical results the analysis of the mean square end to end distance yields thus confirming the exact value to seven significant digits

interacting partially directed walks a model of polymer collapse abstract a survey of the results that have been obtained to date on the partially directed walk model of the polymer collapse transition is presented

on existence regularity and uniqueness of thermally coupled incompressible flows in a system of three dimensional pipes abstract we study an initial boundary value problem for time dependent flows of heat conducting viscous incompressible fluids in a system of three dimensional pipes on a time interval 0 t we are motivated by the bounded domain approach with do nothing boundary conditions in terms of the velocity pressure and temperature of the fluid such flows are described by a coupled parabolic system with strong nonlinearities and including the natural boundary conditions for the velocity and temperature of the fluid on the part of the boundary where the fluid is supposed to leave the channel the present analysis is devoted to the proof of the existence regularity and uniqueness of the solution for the problem described above