adsorption of a self avoiding walk a real space renormalisation group study of d 2 3 the author presents a real space renormalisation group calculation for the adsorption of a single self avoiding walk in d dimensions at a d 1 dimensional impenetrable wall the case d 3 is used to check the method for d 2 the result for the crossover exponent phi gives phi 0 55 or 0 15 this in connection with enumerations of ishinabe 1982 clearly rules out de gennes 1972 1976 1977 conjecture phi 1 v also for the case d 2 for d 2 cells up to 6 6 are used for d 3 cells up to 3 3 3

asymptotic behavior of solutions to nonlinear volterra integral equations in section 2 we establish the existence of solutions to 1 1 and to related equations theorems 2 1 and 2 3 in contrast with previous papers e g 7 we do not assume that a is m accretive instead we require a to satisfy a weaker cone condition certain other facts are also included in this section throughout this paper we also assume that b is completely positive 6 in the third section we present two mean ergodic theorems for the solutions of 1 1 in hilbert space the first theorem theorem 3 1 deals with weak almost convergence and the second theorem 3 4 with strong almost convergence these results improve upon those of 1 and include the known results for nonlinear semigroups 191 it is important to establish almost convergence because it then follows that the solution is summed by every strongly regular kernel corollaries 3 3 and 3 5 a weak convergence criterion corollary 3 2 is also obtained if the banach space x is reflexive and strictly convex and 0 e r a then

asymptotic behavior of solutions of nonlinear volterra equations and mean points abstract in this paper we study the asymptotic behavior of solutions to a nonlinear volterra equation in banach spaces first we deal with the mean point concerning an invariant mean on 0 for the unbounded behavior of solutions in a reflexive banach space using the mean point we obtain the weak and strong convergences for the unbounded behavior of solutions in a reflexive and strictly convex banach space and in the dual space which has a frechet differentiable norm respectively

on an analogue of zarantonello s inequality and a nonlinear mean ergodic theorem abstract let x be a uniformly convex banach space c be a bounded closed subset of x and t c x be a contraction then it is shown that for all xi c i 0 i 1 2 n with i 1n i 1 there exists a strictly increasing convex function n of r with n 0 0 such that n t i 1n ixi i 1n itxi 1 i j n i j xi xj txi txj as an application of this inequality the mean ergodic theorem for contractions in uniformly convex banach spaces is proved

critical exponents for numbers of differently anchored polymer chains on fractal lattices with adsorbing boundaries we study the problem of polymer adsorption in a good solvent when the container of the polymer solvent system is taken to be a member of the sierpinski gasket sg family of fractals members of the sg family are enumerated by an integerb 2 b and it is assumed that one side of each sg fractal is an impenetrable adsorbing boundary we calculate the critical exponents 1 11 and s which within the self avoiding walk model saw of the polymer chain are associated with the numbers of all possible saws with one both and no ends anchored to the adsorbing impenetrable boundary respectively by applying the exact renormalization group rg method for 2 b 8 and the monte carlo renormalization group mcrg method for a sequence of fractals with 2 b 80 we obtain specific values for these exponents the obtained results show that all three critical exponents 1 11 and s in both the bulk phase and crossover region are monotonically increasing functions withb we discuss their mutual relations their relations with other critical exponents pertinent to saws on the sg fractals and their possible asymptotic behavior in the limitb when the fractal dimension of the sg fractals approaches the euclidean value 2

surface adsorption of a self avoiding polymer chain on a family of finitely ramified fractals we study the surface adsorption of a flexible self avoiding polymer chain on a family of sierpinski gasket type fractals using the real space renormalization group technique the members of this family are characterized by an integer scale factor b which runs from 2 to infinity the hausdorff dimension of these fractals tends to 2 from below as b to infinity we calculate the crossover exponent phi for the desorption transition exactly for b 2 to 6 for b to infinity we use finite size scaling arguments to show that phi to 1 4

transfer matrix study of the adsorption of a flexible self avoiding polymer chain in two dimensions the authors model the polymer as a self avoiding walk on a semi infinite square lattice with an energy contribution in from each step along the lattice boundary the critical adsorption temperature ta and the ordinary and special surface exponents of the walk above and at the adsorption temperature respectively are estimated with transfer matrix finite size scaling methods for the adsorption temperature they obtain exp in kta 2 041 or 0 002 and for the crossover and susceptibility exponents of the special transition phi sp 0 501 or 0 003 and gamma 1sp 1 454 or 0 004

critical exponents of surface interacting self avoiding walks on a family of truncated n simplex lattices abstract we study the critical behavior of surface interacting self avoiding random walks on a class of truncated simplex lattices which can be labeled by an integer n 3 using the exact renormalization group method we have been able to obtain the exact values of various critical exponents for all values of n up to n 6 we also derived simple formulas which describe the asymptotic behavior of these exponents in the limit of large n n in spite of the fact that the coordination number of the lattice tends to infinity in this limit we found that most of the studied critical exponents approach certain finite values which differ from corresponding values for simple random walks without self avoiding walk constraint

the direct extension of admm for multi block convex minimization problems is not necessarily convergent the alternating direction method of multipliers admm is now widely used in many fields and its convergence was proved when two blocks of variables are alternatively updated it is strongly desirable and practically valuable to extend the admm directly to the case of a multi block convex minimization problem where its objective function is the sum of more than two separable convex functions however the convergence of this extension has been missing for a long time neither an affirmative convergence proof nor an example showing its divergence is known in the literature in this paper we give a negative answer to this long standing open question the direct extension of admm is not necessarily convergent we present a sufficient condition to ensure the convergence of the direct extension of admm and give an example to show its divergence

renormalisation on sierpinski type fractals presents a family of deterministic fractals which generalise the d dimensional sierpinski gaskets and establishes their order of ramification and their fractal and spectral dimensions random walks on these fractals are renormalisable and lead to rational not necessarily polynomial mapping

crossover exponent for piecewise directed walk adsorption on sierpinski fractals we study the problem of critical adsorption of piecewise directed random walks on a boundary of fractal lattices that belong to the sierpinski gasket family by applying the exact real space renormalization group method we calculate the crossover exponent associated with the number of adsorbed steps for the complete fractal family we demonstrate that our results are very close to the results obtained for ordinary self avoiding walk and discuss the asymptotic behaviour of at the fractal to euclidean lattice crossover

self avoiding walks on fractals studied by the monte carlo renormalization group the authors have applied the monte carlo renormalization group mcrg method to study the problem of self avoiding walks on the sierpinski gasket family of fractals each member of the family is labelled by an integer b 2 or b or infinity and when b to infinity both the fractal df and spectral ds dimension approach their euclidean value 2 they have calculated the critical exponent nu associated with the mean square end to end distance up to b 80 their mcrg results deviate at most 0 03 from the available exact results for 2 or b or 9 the obtained data show clearly that nu monotonically decreases with b and crosses the euclidean value nu 3 4 at b approximately 27 that is before entering the fractal to euclidean crossover region that occurs in the limit b to infinity

critical exponents of the self avoiding walks on a family of finitely ramified fractals the authors have studied the self avoiding walks saw on a family of finitely ramified fractals the first member b 2 of the family is the two dimensional sierpinski gasket while the last member b infinity appears to be a wedge of the homogeneous triangular lattice by means of the exact renormalisation group transformations they have calculated the critical exponents alpha nu and gamma and the connectivity constant mu of saw on each member of a sequence 2 or b or 8 of the studied fractal family the obtained exact results are compared with the recent phenomenological proposals and with the results believed to be exact in the case of a homogeneous two dimensional lattice

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

asymptotic behavior of solutions of nonlinear volterra equations 0 recently baillon and clement 3 established mean ergodic theorems for solutions of nonlinear volterra equations in hilbert space in the present paper we study the asymptotic behavior of solutions of 1 1 by using nonlinear ergodic theorems the first mean ergodic theorem was given by baillon l and it was improved by several authors 3 8 141 163

almost convergence of solutions of nonlinear volterra equations in banach space a l aide d un theoreme ergodique on etudie le comportement asymptotique des solutions des problemes aux valeurs initiales dans des espaces de banach on applique le resultat obtenu a l etude du comportement asymptotique des solutions des equations de volterra non lineaires dans des espaces de banach

critical behaviour of two interacting linear polymer chains exact results for a state of interpenetration of chains on a fractal lattice a model of two interacting chemically different linear polymer chains on a fractal lattice is proposed to study segregation and interpenetration of chains using the real space renormalization group transformation the value of the contact exponent at the point of the interpenetration transition is calculated exactly for the 2d sierpinski gasket

latent variable graphical model selection via convex optimization suppose we observe samples of a subset of a collection of random variables no additional information is provided about the number of latent variables nor of the relationship between the latent and observed variables is it possible to discover the number of latent components and to learn a statistical model over the entire collection of variables we address this question in the setting in which the latent and observed variables are jointly gaussian with the conditional statistics of the observed variables conditioned on the latent variables being specified by a graphical model as a first step we give natural conditions under which such latent variable gaussian graphical models are identifiable given marginal statistics of only the observed variables essentially these conditions require that the conditional graphical model among the observed variables is sparse while the effect of the latent variables is spread out over most of the observed variables next we propose a tractable convex program based on regularized maximum likelihood for model selection in this latent variable setting the regularizer uses both the ell 1 norm and the nuclear norm our modeling framework can be viewed as a combination of dimensionality reduction to identify latent variables and graphical modeling to capture remaining statistical structure not attributable to the latent variables and it consistently estimates both the number of latent components and the conditional graphical model structure among the observed variables these results are applicable in the high dimensional setting in which the number of latent observed variables grows with the number of samples of the observed variables the geometric properties of the algebraic varieties of sparse matrices and of low rank matrices play an important role in our analysis

monte carlo renormalization group study of crosslinked polymer chains on fractals we study the problem of two crosslinked polymer chains in a good solvent modelled by two mutually crossing self avoiding walks situated on fractals that belong to the sierpinski gasket sg family whose members are labelled by an integer b by applying the monte carlo renormalization group mcrg method we calculate the critical exponent y associated with the number of crossings of the two self avoiding walk paths for a sequence of sg fractals with for the problem under study we find that our mcrg approach provides results that are virtually rigorous that is results with exceptionally small deviations at most 0 07 from the available exact renormalization group results we discuss our set of mcrg data for y as a function of the fractal parameter b and compare its behaviour with the finite size scaling predictions

recovering low rank and sparse components of matrices from incomplete and noisy observations many problems can be characterized by the task of recovering the low rank and sparse components of a given matrix recently it was discovered that this nondeterministic polynomial time hard np hard task can be well accomplished both theoretically and numerically via heuristically solving a convex relaxation problem where the widely acknowledged nuclear norm and l 1 norm are utilized to induce low rank and sparsity this paper studies the recovery task in the general settings that only a fraction of entries of the matrix can be observed and the observation is corrupted by both impulsive and gaussian noise we show that the resulting model falls into the applicable scope of the classical augmented lagrangian method moreover the separable structure of the new model enables us to solve the involved subproblems more efficiently by splitting the augmented lagrangian function hence some splitting numerical algorithms are developed for solving the new recovery model some preliminary numerical experiments verify that these augmented lagrangian based splitting algorithms are easily implementable and surprisingly efficient for tackling the new recovery model

alternating direction method with gaussian back substitution for separable convex programming we consider the linearly constrained separable convex minimization problem whose objective function is separable into m individual convex functions with nonoverlapping variables a douglas rachford alternating direction method of multipliers adm has been well studied in the literature for the special case of m 2 but the convergence of extending adm to the general case of m ge 3 is still open in this paper we show that the straightforward extension of adm is valid for the general case of m ge 3 if it is combined with a gaussian back substitution procedure the resulting adm with gaussian back substitution is a novel approach towards the extension of adm from m 2 to m ge 3 and its algorithmic framework is new in the literature for the adm with gaussian back substitution we prove its convergence via the analytic framework of contractive type methods and we show its numerical efficiency by some application problems

non equilibrium critical behaviour on fractal lattices a simple one component surface reaction model exhibiting a critical point has been studied on deterministic fractal lattices with fractal dimension 1 df 2 results from time dependent simulations show a change in the value of the critical exponents with df two of the exponents interpolate very nicely between the values in one and two dimensions

convergence rate analysis for the alternating direction method of multipliers with a substitution procedure for separable convex programming recently in he et al he bs tao m yuan xm 2012 alternating direction method with gaussian back substitution for separable convex programming siam j optim 22 2 313 340 we have showed the first possibility of combining the douglas rachford alternating direction method of multipliers admm with a gaussian back substitution procedure for solving a convex minimization model with a general separable structure this paper is a further study on this theme we first derive a general algorithmic framework to combine admm with either a forward or backward substitution procedure then we show that convergence of this framework can be easily proved from the contraction perspective and its local linear convergence rate is provable if certain error bound condition is assumed without such an error bound assumption we can estimate its worst case convergence rate measured by the iteration complexity

convergent prediction correction based admm for multi block separable convex programming abstract the direct extension of the classic alternating direction method with multipliers admme to the multi block separable convex optimization problem is not necessarily convergent though it often performs very well in practice in order to preserve the numerical advantages of admme and obtain convergence many modified admm were proposed by correcting the output of admme or employing proximal terms to solve inexactly the subproblems in admme in this paper we present an efficient prediction correction based admm pcb admm to solve the multi block separable convex minimization model the prediction step takes a special block coordinate descent bcd cycle to update the variable blocks then the correction step corrects the output slightly by computing a convex combination of two points from the prediction step and previous iteration the convergence property is obtained by using the variational inequality the numerical experiments illustrate effectiveness of the proposed pcb admm to solve the quadratic semidefinite programming and image decomposition

asymptotic behaviour of almost nonexpansive sequences in a hilbert space abstract the results in 9 11 12 for nonexpansive sequences are generalized to almost nonexpansive sequences in a hilbert space and using this notion a direct proof of a result of h brezis and f e browder 6 is given

on the o 1 n convergence rate of the douglas rachford alternating direction method alternating direction methods adms have been well studied in the literature and they have found many efficient applications in various fields in this note we focus on the douglas rachford adm scheme proposed by glowinski and marrocco and we aim at providing a simple approach to estimating its convergence rate in terms of the iteration number the linearized version of this adm scheme which is known as the split inexact uzawa method in the image processing literature is also discussed

asymptotic behavior of the solutions of a class of functional differential equations remarks on a related volterra equation on etudie la convergence asymptotique forte quand t des solutions de u t au t g u t f t t 0 u 0 u 0 a est un operateur monotone maximal defini sur un sous ensemble d a contenu dans un espace de hilbert reel h u 0 d a f l 1 0 h et g est une application donnee

self interacting polymer chains terminally anchored to adsorbing surfaces of three dimensional fractal lattices abstract we have studied the adsorption problem of self attracting linear polymers modeled by self avoiding walks saws situated on three dimensional fractal structures exemplified by 3d sierpinski gasket sg family of fractals as containers of a poor solvent members of sg family are enumerated by an integer b b 2 and it is assumed that one side of each sg fractal is an impenetrable adsorbing surface we calculate the critical exponents 1 11 and s which are related to the numbers of all possible saws with one both and no ends anchored to the adsorbing boundary respectively by applying the exact renormalization group rg method for the first three members of the sg fractal family b 2 3 and 4 we have obtained specific values of these exponents for chain and globular polymer phase we discuss their mutual relations and relations with corresponding values pertinent to extended polymer chain phase

on the critical behaviour of a surface interacting linear polymer chain abstract a method based on a real space renormalization group transformation is developed to describe the critical behaviour of a surface interacting linear flexible polymer chain represented by a self avoiding walk it is shown that a lattice model based on a central rule in which the starting point of the walk and the surface are taken to be in the middle of one cell provides a suitable framework to study both the penetrable and impenetrable surfaces in contrast to this a method based on a corner rule in which the starting point of the walk and the surface are fixed to be one corner of a cell cannot describe the behaviour of a chain interacting with an impenetrable surface the value of crossover exponent found by us for a square lattice are in agreement with those expected to be exact for both the cases

critical behavior of two interacting linear polymer chains in a good solvent a model of two interacting chemically different linear polymer chains is solved exactly using the real space renormalization group transformation on a family of sierpinski gasket type fractals and on a truncated 4 simplex lattice the members of the family of the sierpinski gasket type fractals are characterized by an integer scale factorb which runs from 2 to the hausdorff dimensiond f of these fractals tends to 2 from below asb we calculate the contact exponenty for the transition from the state of segregation to a state in which the two chains are entangled forb 2 5 using arguments based on the finite size scaling theory we show that forb y 2 v b d f wherev is the end toend distance exponent of a chain for a truncated 4 simplex lattice it is shown that the system of two chains either remains in a state in which these chains are intermingled in such a way that they cannot be told apart in the sense that the chemical difference between the polymer chains completely drop out of the thermodynamics of the system or in a state in which they are either zipped or entangled we show the region of existence of these different phases separated by tricritical lines the value of the contact exponenty is calculated at the tricritical points

critical adsorption of random walks on fractal lattices with uniform coordination number abstract we study the problem of adsorption of random walks on a boundary of fractal lattices that have uniform coordination numbers more specifically for a suitable gaussian model situated on the sierpinski gasket fractals with an interacting wall we analyze critical properties using the renormalization group approach in this way we have found exact expressions for a set of pertinent critical exponents in particular we have demonstrated that the crossover critical exponent associated with the number of adsorbed monomers can be expressed as simple combination of only three quantities the end to end distance critical exponent the substratum fractal dimension and the adsorbing boundary fractal dimension

asymptotic properties of solutions of time dependent volterra integral equations abstract the asymptotic properties of the bounded solutions of a time dependent nonlinear volterra equation are studied in a hilbert space an application of physical interest is also given

on the linear convergence of the alternating direction method of multipliers we analyze the convergence rate of the alternating direction method of multipliers admm for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints previous analysis of the admm typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks moreover there has been no rate of convergence analysis for the admm without strong convexity in the objective function in this paper we establish the global r linear convergence of the admm for minimizing the sum of any number of convex separable functions assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping and a nonsmooth ell 1 l1 regularizer this result implies the linear convergence of the admm for contemporary applications such as lasso without assuming strong convexity of the objective function

asymptotic behaviour of quasi autonomous dissipative systems in hilbert spaces abstract by suitably modifying the methods used by b djafari rouhani ergodic theorems for non expansive sequences in hilbert spaces and related problems part i pp 1 76 thesis yale university and a new proof of the weak convergence theorems for non expansive sequences and curves in hilbert spaces preprint 1984 and by b djafari rouhani and s kakutani ergodic theorems for non expansive non linear operators in a hilbert space preprint 1984 we study the asymptotic behaviour of the quasi autonomous dissipative system du dt au where a is a monotone operator in a hilbert space h and l 1 0 h