atmospheres of polygons and knotted polygons in this paper we define two statistics a and a the positive and negative atmospheres of a lattice polygon of fixed length n these statistics have the property that a a pn 2 pn where pn is the number of polygons of length n counted modulo translations we use the pivot algorithm to sample polygons and to compute the corresponding average atmospheres using these data we directly estimate the growth constants of polygons in two and three dimensions we find that where the error bars are 67 confidence intervals and the second error bar in the three dimensional estimate of is an estimated systematic error we also compute atmospheres of polygons of fixed knot type k sampled by the bfacf algorithm we discuss the implications of our results and show that different knot types have atmospheres which behave dramatically differently at small values of n

extended sufficient semilocal convergence for the secant method we establish new sufficient convergence conditions for the secant method to a locally unique solution of a nonlinear equation in a banach space using our new concept of recurrent functions and combining lipschitz and center lipschitz conditions on the divided difference operator we obtain a new semilocal convergence analysis of the secant method moreover our sufficient convergence conditions expand the applicability of the secant method in cases not covered before dennis 1971 9 hernandez et al 2005 8 laasonen 1969 15 ortega and rheinboldt 1970 11 potra 1982 5 potra 1985 7 schmidt 1978 18 yamamoto 1987 12 wolfe 1978 19 numerical examples are also provided in this study

localization criteria for anderson models on locally finite graphs we prove spectral and dynamical localization for anderson models on locally finite graphs using the fractional moment method our theorems extend earlier results on localization for the anderson model on d we establish geometric assumptions for the underlying graph such that localization can be proven in the case of sufficiently large disorder

on the local convergence of a newton kurchatov type method for non differentiable operators abstract by means of a nice idea a newton kurchatov type iterative process is constructed for solving nonlinear equations in banach spaces we analyze the local convergence of this iterative process this study have an important and novel feature since it is applicable to non differentiable operators so far most of the local convergence results considered by other authors may apply only to differentiable operators due to the conditions that are required on the solution of the nonlinear equation

a method for finding sharp error bounds for newton s method under the kantorovic assumptions this paper gives a method for finding sharpa posteriori error bounds for newton s method under the assumptions of kantorovich s theorem on the basis of this method new error bounds are derived and comparison is made among the known bounds of dennis 2 doring 4 gragg tapia 5 kantorovich 6 7 kornstaedt 9 lancaster 10 miel 11 13 moret 14 ostrowski 17 18 potra 19 and potra ptak 20

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

the size and number of rings on the square lattice the authors have calculated the number of self avoiding polygons on the square lattice to 56 steps and the caliper size to 54 steps analysis of the generating function permits estimates of the connective constant mu 2 638 1585 or 10 6 and the critical exponents alpha 0 500 06 or 0 000 06 and v 0 753 or 0 007 the singularity structure of the polygon generating function is found to be consistent with a correction to the scaling exponent delta 1 5 as predicted by nienhuis 1982 1984 the confluent part however maps into the additive analytic term due to the value of the exponent alpha

critical exponents hyperscaling and universal amplitude ratios for two and three dimensional self avoiding walks we make a high precision monte carlo study of two and three dimensional self avoiding walks saws of length up to 80 000 steps using the pivot algorithm and the karp luby algorithm we study the critical exponents v and 2 triangle sub 4 gamma as well as several universal amplitude ratios in particular we make an extremely sensitive test of the hyperscaling relation dv 2 triangle sub 4 gamma in two dimensions we confirm the predicted exponent v 3 4 and the hyperscaling relation we estimate the universal ratios 0 14026 0 00007 0 43961 0 00034 and psi 0 66296 0 00043 68 confidence limits in three dimensions we estimate v 0 5877 0 0006 with a correction to scaling exponent triangle sub 1 0 56 0 03 subjective 68 confidence limits this value for v agrees excellently with the field theoretic renormalization group prediction but there is some discrepancy for triangle 1 earlier monte carlo estimates of v which were approximately 0 592 are now seen to be biased by corrections to scaling we estimate the universal ratios 0 1599 0 0002 and psi 0 2471 0 0003 since psi 0 hyperscaling holds the approach to psi is from above contrary to the prediction of the two parameter renormalization group theory we critically reexamine this theory and explain where the error lies in an appendix we prove rigorously modulo some standard scalingmore assumptions the hyperscaling relation dv 2 triangle sub 4 gamma for two dimensional saws less

on the local convergence of a family of two step iterative methods for solving nonlinear equations a local convergence analysis for a generalized family of two step secant like methods with frozen operator for solving nonlinear equations is presented unifying earlier methods such as secant s newton chebyshev like steffensen and other new variants the family of iterative schemes is built up where a profound and clear study of the computational efficiency is also carried out numerical examples and an application using multiple precision and a stopping criterion are implemented without using any known root finally a study comparing the order efficiency and elapsed time of the methods suggested supports the theoretical results claimed

a semilocal convergence analysis for directional newton methods a semilocal convergence analysis for directional newton methods in n variables is provided in this study using weaker hypotheses than in the elegant related work by y levin and a ben israel and introducing the center lipschitz condition we provide under the same computational cost as in levin and ben israel a semilocal convergence analysis with the following advantages weaker convergence conditions larger convergence domain finer error estimates on the distances involved and an at least as precise information on the location of the zero of the function a numerical example where our results apply to solve an equation but not the ones in levin and ben israel is also provided in this study

a variant of the newton kantorovich theorem for nonlinear integral equations of mixed hammerstein type abstract we study nonlinear integral equations of mixed hammerstein type using newton s method as follows we investigate the theoretical significance of newton s method to draw conclusions about the existence and uniqueness of solutions of these equations after that we approximate the solutions of a particular nonlinear integral equation by newton s method for this we use the majorant principle which is based on the concept of majorizing sequence given by kantorovich and milder convergence conditions than those of kantorovich actually we prove a semilocal convergence theorem which is applicable to situations where kantorovich s theorem is not

mean field critical behaviour for correlation length for percolation in high dimensions extending the method of 27 we prove that the corrlation length of independent bond percolation models exhibits mean field type critical behaviour i e p p c p 1 2 asp p c in two situations i for nearest neighbour independent bond percolation models on ad dimensional hypercubic lattice d withd sufficiently large and ii for a class of spread out independent bond percolation models which are believed to belong to the same universality class as the nearest neighbour model in more than six dimensions the proof is based on and extends a method developed in 27 where it was used to prove the triangle condition and hence mean field behaviour of the critical exponents and 2 for the above two cases

the majorant method in the theory of newton kantorovich approximations and generalized lipschitz conditions we provide a semilocal as well as a local convergence analysis for newton s and modified newton s methods in order to approximate a locally unique solution of a nonlinear equation in a banach space setting we use more precise majorizing sequences than in earlier studies such as appell et al 1997 appell et al 1991 argyros 2004 argyros and hilout 2009 kantorovich and akilov 1982 ortega and rheinboldt 1970 and generalized lipschitz continuity conditions our sufficient convergence conditions are weaker than before and our convergence analysis is tighter special cases and numerical examples are also given in this study

newton kantorovich theorem for a family of modified halley s method under holder continuity conditions in banach space abstract in this study under p holder continuity conditions a family of new modified halley s method with 2 p order is proposed in banach space which is used to solve the nonlinear operator equations the newton kantorovich convergence theorem for the family of new modified halley s method is established under holder continuity conditions the error estimate also is gotten finally two examples are provided to show the application of our theorem

bias remedy least mean square equation error algorithm for iir parameter recursive estimation in the area of infinite impulse response iir system identification and adaptive filtering the equation error algorithms used for recursive estimation of the plant parameters are well known for their good convergence properties however these algorithms give biased parameter estimates in the presence of measurement noise a new algorithm is proposed on the basis of the least mean square equation error lmsee algorithm which manages to remedy the bias while retaining the parameter stability the so called bias remedy least mean square equation error brle algorithm has a simple form the compatibility of the concept of bias remedy with the stability requirement for the convergence procedure is supported by a practically meaningful theorem the behavior of the brle has been examined extensively in a series of computer simulations

high precision canonical monte carlo determination of the growth constant of square lattice trees exponent in this paper we show that l and u can be determined to high accuracy by using a canonical monte carlo algorithm we find thatl55 143960 0025 u51 01460 022 where the error bars are a combined 95 statistical confidence interval and an estimated systematic error due to uncertainties in modeling corrections to scaling if one assumes the exact value u51 and then determines l then the above estimate improves to l55 143 3960 000 72 in addition we also determine the longest path exponent r and the metric exponent n from our data r50 7400060 00062 n50 643760 0035 with error bars similarly a combined 95 statistical confidence interval and an estimate of the systematic error

matching measure benjamini schramm convergence and the monomer dimer free energy we define the matching measure of a lattice l as the spectral measure of the tree of self avoiding walks in l we connect this invariant to the monomer dimer partition function of a sequence of finite graphs converging to l this allows us to express the monomer dimer free energy of l in terms of the matching measure exploiting an analytic advantage of the matching measure over the mayer series then leads to new rigorous bounds on the monomer dimer free energies of various euclidean lattices while our estimates use only the computational data given in previous papers they improve the known bounds significantly

a note on mean field behavior for self avoiding walk on branching planes we consider the critical behavior of the susceptibility of the self avoiding walk on the grapht z wheret is a bethe lattice with degreek andz is the one dimensional lattice by directly estimating the two point function using a method of grimmett and newman we show that the bubble condition is satisfied whenk 2 and therefore the critical exponent associated with the susceptibility equals 1

extending the newton kantorovich hypothesis for solving equations the famous newton kantorovich hypothesis kantorovich and akilov 1982 3 argyros 2007 2 argyros and hilout 2009 7 has been used for a long time as a sufficient condition for the convergence of newton s method to a solution of an equation in connection with the lipschitz continuity of the frechet derivative of the operator involved here using lipschitz and center lipschitz conditions and our new idea of recurrent functions we show that the newton kantorovich hypothesis can be weakened under the same information moreover the error bounds are tighter than the corresponding ones given by the dominating newton kantorovich theorem argyros 1998 1 2 7 ezquerro and hernandez 2002 11 3 proinov 2009 2010 16 17 numerical examples including a nonlinear integral equation of chandrasekhar type chandrasekhar 1960 9 as well as a two boundary value problem with a green s kernel argyros 2007 2 are also provided in this study

kantorovich s type theorems for systems of equations with constant rank derivatives the famous newton kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of newton s method to a solution of an equation here we present a kantorovich type convergence analysis for the gauss newton s method which improves the result in w m hauszler a kantorovich type convergence analysis for the gauss newton method numer math 48 1986 119 125 and extends the main theorem in i k argyros on the newton kantorovich hypothesis for solving equations j comput appl math 169 2004 315 332 furthermore the radius of convergence ball is also obtained

on the newton kantorovich hypothesis for solving equations the famous newton kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of newton s method to a solution of an equation in connection with the lipschitz continuity of the frechet derivative of the operator involved here using lipschitz and center lipschitz conditions we show that the newton kantorovich hypothesis can be weakened the error bounds obtained under our semilocal convergence result are more precise than the corresponding ones given by the dominating newton kantorovich theorem

scaling of self avoiding walks in high dimensions we examine self avoiding walks in dimensions 4 to 8 using high precision monte carlo simulations up to length n 16 384 providing the first such results in dimensions d 4 on which we concentrate our analysis we analyse the scaling behaviour of the partition function and the statistics of nearest neighbour contacts as well as the average geometric size of the walks and compare our results to 1 d expansions and to excellent rigorous bounds that exist in particular we obtain precise values for the connective constants 5 8 838 544 3 6 10 878 094 4 7 12 902 817 3 8 14 919 257 2 and give a revised estimate of 4 6 774 043 5 all of these are by at least one order of magnitude more accurate than those previously given from other approaches in d 4 and all approaches in d 4 our results are consistent with most theoretical predictions though in d 5 we find clear evidence of anomalous n 1 2 corrections for the scaling of the geometric size of the walks which we understand as a non analytic correction to scaling of the general form n 4 d 2 not present in pure gaussian random walks

correction to scaling exponents and critical properties of the n vector model with dimensionality 4 a perturbation calculation is given which implies that the susceptibility of the five and six dimensional n vector models can be written x approximately at 1 1 bt12 and x approximately at 1 1 bt in t respectively independent of n for n 0 and 1 it is shown that series analysis techniques can extract the correction to scaling exponents and that estimates of the critical temperatures and critical amplitudes can also be obtained the correction to scaling exponents found are in agreement with those known to exist in the case of the spherical model

newton methods on banach spaces with a convergence structure and applications abstract in this study we use inexact newton like methods to find solutions of nonlinear operator equations on banach spaces with a convergence structure our technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered banach space in this way the metric properties of the examined problem can be analyzed more precisely moreover this approach allows us to derive from the same theorem on the one hand semilocal results of kantorovich type and on the other hand global results based on monotonicity considerations by imposing very general lipschitz like conditions on the operators involved on the one hand we cover a wider range of problems and on the other hand by choosing our operators appropriately we can find sharper error bounds on the distances involved than before finally our methods are used to solve integral equations that cannot be solved with existing methods

the shape of self avoiding walks an exhaustive analysis of the shape properties of discrete self avoiding random walks is presented the dependence of the main parameters of the probability distributions from the length of the walk and the differences between these distributions for conventional and self avoiding walks are studied by means of high precision monte carlo simulations it is shown that the characteristics of the shape of self avoiding random walks when regarded as a function of the walk length present the expected asymptotic behaviour the differences with conventional random walks depend upon the observable considered the probability distributions of the principal inertia eigenvalues of self avoiding walks spread around the most probable value more widely than the corresponding distributions for unrestricted walks while in the cases of the asphericity or ratios of inertia eigenvalues the distributions for self avoiding walks are somewhat more peaked than their counterparts for conventional random walks analytical expressions for the probability distributions of inertia moment ratios and the two dimensional asphericity are given for common random walks there is an excellent agreement between these analytical distributions and the monte carlo data in this work it is established that this concordance is maintained if the same analytical distributions are applied to the self avoiding case this means that within the precision of the simulations the functional form of the mentioned distributions does not vary when passing from conventional to self avoiding walks

a newton kantorvich theorem for equations involving m frechet differentiable operators and applications in radiative transfer abstract in this study we approximate a locally unique solution of a nonlinear operator equation in banach space using newton s method a complete error analysis showing the quadratic convergence of our method is also given our new theorem uses lipschitz or holder continuity assumptions on m frechet differentiable operators where m 2 is a positive integer a numerical example is given to show that our results provide a better information on the location of the solution as well as finer error bounds on the distances involved than earlier results a second numerical example shows how to solve a nonlinear integral equation appearing in radiative transfer

self avoiding walk enumeration via the lace expansion we introduce a new method for the enumeration of self avoiding walks based on the lace expansion we also introduce an algorithmic improvement called the two step method for self avoiding walk enumeration problems we obtain significant extensions of existing series on the cubic and hypercubic lattices in all dimensions d 3 we enumerate 32 step self avoiding polygons in d 3 26 step self avoiding polygons in d 4 30 step self avoiding walks in d 3 and 24 step self avoiding walks and polygons in all dimensions d 4 we analyze these series to obtain estimates for the connective constant and various critical exponents and amplitudes in dimensions 3 d 8 we also provide major extensions of 1 d expansions for the connective constant and for two critical amplitudes

third order convergence theorem by using majorizing function for a modified newton method in banach space abstract following the ideas of frontini and sormani we present a modified newton method in banach space which is used to solve the nonlinear operator equation we establish the newton kantorovich convergence theorem for the modified newton method with third order convergence in banach space by using majorizing function we also get the error estimate finally two examples are provided to show the application of our theorem

weak convergence conditions for inexact newton type methods abstract we establish a new semilocal convergence results for inexact newton type methods for approximating a locally unique solution of a nonlinear equation in a banach spaces setting we show that our sufficient convergence conditions are weaker and the estimates of error bounds are tighter in some cases than in earlier works 15 31 special cases and numerical examples are also provided in this study

majorizing sequences for iterative methods we provide convergence results for very general majorizing sequences of iterative methods using our new concept of recurrent functions we unify the semilocal convergence analysis of newton type methods ntm under more general lipschitz type conditions we present two very general majorizing sequences and we extend the applicability of ntm using the same information before chen and yamamoto 1989 13 deuflhard 2004 16 kantorovich and akilov 1982 19 miel 1979 20 miel 1980 21 and rheinboldt 1968 30 applications special cases and examples are also provided in this study to justify the theoretical results of our new approach

expanding the applicability of newton s method using smale s α theory we present a tighter convergence analysis than earlier studies such as in cianciaruso 2007 guo 2007 shen and li 2010 smale 1986 1987 wang and zhao 1995 wang 1999 wang and han 1990 of newton s method using smale s a theory by introducing the notion of the center c 0 condition in particular in the semilocal convergence case we show that if the center c 0 condition is smaller than the c condition then the new majorizing sequence is tighter than the old majorizing sequence the new convergence criteria are weaker than the older convergence criteria furthermore in the local convergence case we obtain a larger radius of convergence and tighter error estimates on the distances involved these improvements are obtained under the same computational cost numerical examples and applications are also provided in this study to show that the older results cannot apply but the new results apply to solve equations

on newton s method for solving equations containing frechet differentiable operators of order at least two we provide sufficient conditions for the semilocal convergence of newton s method to a locally unique solution of a nonlinear operator equation containing operators that are frechet differentiable of order at least two in a banach space setting numerical examples are also provided to show that our results apply to solve nonlinear equations in cases earlier ones cannot j m gutierrez a new semilocal convergence theorem for newton s method j comput appl math 79 1997 131 145 z huang a note of kantorovich theorem for newton iteration j comput appl math 47 1993 211 217 f a potra sharp error bounds for a class of newton like methods libertas mathematica 5 1985 71 84

a steffensen s type method in banach spaces with applications on boundary value problems in this paper a modified steffensen s type iterative scheme for the numerical solution of a system of nonlinear equations is studied two convergence theorems are presented the numerical solution of boundary value problems by the multiple shooting method using the proposed iterative scheme is analyzed

general 2 2 system of nonlinear integral equations and its approximate solution abstract in this note we consider a general 2 2 system of nonlinear volterra type integral equations the modified newton method modified nm is used to reduce the nonlinear problems into 2 2 linear system of algebraic integral equations of volterra type the latter equation is solved by discretization method nystrom method with gauss legendre quadrature is applied for the kernel integrals and newton forwarded interpolation formula is used for finding values of unknown functions at the selected node points existence and uniqueness solution of the problems are proved and accuracy of the quadrature formula together with convergence of the proposed method are obtained finally numerical examples are provided to show the validity and efficiency of the method presented numerical results reveal that the proposed methods is efficient and accurate comparisons with other methods for the same problem are also presented

algebraic techniques for enumerating self avoiding walks on the square lattice the authors describe a new algebraic technique for enumerating self avoiding walks on the rectangular lattice the computational complexity of enumerating walks of n steps is of order 3n 4 times a polynomial in n and so the approach is greatly superior to direct counting techniques they have enumerated walks of up to 39 steps as a consequence they are able to accurately estimate the critical point critical exponent and critical amplitude

self testing algorithms for self avoiding walks we present a markov chain monte carlo algorithm for almost uniformly generating and approximately counting self avoiding walks in rectangular lattices zd these are classical problems that arise for example in the study of long polymer chains while there are a number of monte carlo algorithms used to solve these problems in practice these are heuristic and their correctness relies on unproven conjectures in contrast our algorithm is shown rigorously to produce answers with specified accuracy and confidence only the efficiency of the algorithm relies on a widely believed conjecture and a novel feature is that this conjecture can be tested as the algorithm proceeds with this self testing feature incorporated the algorithm has polynomially bounded running time and is completely reliable in the sense that it either outputs answers that are guaranteed to be within the specified accuracy and confidence bounds or finds a counter example to the conjecture

weak convergence conditions for the newton s method in banach space using general majorizing sequences using the majorizing principle we develop sufficient convergence conditions for the semilocal convergence of newton s method in a banach space setting the involved majorizing sequences are very general and as special cases include most relevant sequences from the published literature our convergence conditions can be weaker and our error analysis can be finer than before

lower bound on the connective constant for square lattice self avoiding walks by enumerating irreducible bridges exactly up to 40 steps and by obtaining a lower bound to the number of bridges with less than 125 steps a lower bound for the connective constant for square lattice self avoiding walks of 2 62 is obtained

new lower bounds on the self avoiding walk connective constant we give an elementary new method for obtaining rigorous lower bounds on the connective constant for self avoiding walks on the hypercubic lattice d the method is based on loop erasure and restoration and does not require exact enumeration data our bounds are best for highd and in fact agree with the first four terms of the 1 d expansion for the connective constant the bounds are the best to date for dimensionsd 3 but do not produce good results in two dimensions ford 3 4 5 and 6 respectively our lower bound is within 2 4 0 43 0 12 and 0 044 of the value estimated by series extrapolation

on the local convergence of fast two step newton like methods for solving nonlinear equations a local convergence analysis is presented for a fast two step newton like method tsnlm for solving nonlinear equations in a banach space setting the tsnlm unifies earlier methods such as newton s secant newton like chebyshev secant chebyshev newton steffensen stirling s and other single or multistep methods numerical examples and a comparative study of these methods validating our theoretical results are also given in the concluding section of this paper

on a class of newton like methods for solving nonlinear equations we provide a semilocal convergence analysis for a certain class of newton like methods considered also in i k argyros a unifying local semilocal convergence analysis and applications for two point newton like methods in banach space j math anal appl 298 2004 374 397 i k argyros computational theory of iterative methods in c k chui l wuytack eds series studies in computational mathematics vol 15 elsevier publ co new york usa 2007 j e dennis toward a unified convergence theory for newton like methods in l b rall ed nonlinear functional analysis and applications academic press new york 1971 in order to approximate a locally unique solution of an equation in a banach space using a combination of lipschitz and center lipschitz conditions instead of only lipschitz conditions f a potra sharp error bounds for a class of newton like methods libertas math 5 1985 71 84 we provide an analysis with the following advantages over the work in f a potra sharp error bounds for a class of newton like methods libertas math 5 1985 71 84 which improved the works in w e bosarge p l falb a multipoint method of third order j optimiz theory appl 4 1969 156 166 w e bosarge p l falb infinite dimensional multipoint methods and the solution of two point boundary value problems numer math 14 1970 264 286 j e dennis on the kantorovich hypothesis for newton s method siam j numer anal 6 3 1969 493 507 j e dennis toward a unified convergence theory for newton like methods in l b rall ed nonlinear functional analysis and applications academic press new york 1971 h j kornstaedt ein allgemeiner konvergenzstaz fu r verscha rfte newton verfahrem in isnm vol 28 birkhau ser verlag basel and stuttgart 1975 pp 53 69 p laasonen ein uberquadratisch konvergenter iterativer algorithmus ann acad sci fenn ser i 450 1969 1 10 f a potra on a modified secant method l analyse numerique et la theorie de l approximation 8 2 1979 203 214 f a potra an application of the induction method of v ptak to the study of regula falsi aplikace matematiky 26 1981 111 120 f a potra on the convergence of a class of newton like methods in iterative solution of nonlinear systems of equations in lecture notes in mathematics vol 953 springer verlag new york 1982 f a potra v ptak nondiscrete induction and double step secant method math scand 46 1980 236 250 f a potra v ptak on a class of modified newton processes numer funct anal optim 2 1 1980 107 120 f a potra sharp error bounds for a class of newton like methods libertas math 5 1985 71 84 j w schmidt untere fehlerschranken fur regula falsi verfahren period math hungar 9 3 1978 241 247 j w schmidt h schwetlick ableitungsfreie verfhren mit hoherer konvergenzgeschwindifkeit computing 3 1968 215 226 j f traub iterative methods for the solution of equations prentice hall englewood cliffs new jersey 1964 m a wolfe extended iterative methods for the solution of operator equations numer math 31 1978 153 174 larger convergence domain and weaker sufficient convergence conditions numerical examples further validating the results are also provided

bounds on connective constants for self avoiding walks new lower bounds on the connective constant of the square and simple cubic lattice self avoiding walks are obtained by enumerating a particular subset of self avoiding walks and using a result of kesten the author finds mu sq 2 5680 and mu sc 4 352

marriage of exact enumeration and 1 d expansion methods lattice model of dilute polymers we consider the properties of a self avoiding polymer chain with nearestneighbor contact energy on ad dimensional hypercubic lattice general theoretical arguments enable us to prescribe the exact analytic form of then segment chain partition functionc n and unknown coefficients for chains of up to 11 segments are determined using exact enumeration data ind 2 6 this exact form provides the main ingredient to produce a large n expansion ind 1of the chain free energy through fifth order with the full dependence on the contact energy retained the dependent chain connectivity constant and free energy amplitude are evaluated within thed 1expansion too d 5 our general formulation includes for the first time self avoiding walks neighboravoiding walks theta and collapsed chains as particular limiting cases

self avoiding walk contacts and random walk self intersections in variable dimensionality the average number of nearest neighbor nn contacts m of self avoiding walks saw s on a hypercubic lattice is calculated using direct enumeration and 1 d expansion methods where d is the spatial dimension these calculations are compared with exact analytic determinations for the asymptotic number of random walk rw self intersections in the limit of long chains n ensuremath rightarrow ensuremath infty the number of rw binary ternary etc self intersections is a function of the probability mathit c mathit d mathrm that a rw escapes from the origin to infinity and an accurate tabulation of mathit c mathit d mathrm is given in the dimension range 2d10 we find that the number of saw nn contacts m mathrm mathrm saw has an asymptotic behavior m mathrm mathrm saw ensuremath sim mathit a mathrm ensuremath infty n similar to that for the number of rw self intersections m mathrm mathrm rw ensuremath sim mathit a mathrm rw n as first suggested by domb but the corrections to this leading scaling differ for the rw and saw problems the contact amplitude mathit a mathrm ensuremath infty determined from direct enumeration data and ratio extrapolation exhibits a maximum near d 3 dimensions as does mathit a mathrm rw comparison of the numerical estimates for mathit a mathrm ensuremath infty to the 1 d expansion calculation of mathit a mathrm ensuremath infty shows significant deviation for d5 reflecting the strong fluctuations in contacts that arise in lower dimensions the correction to the scaling exponent mathrm ensuremath delta mathit m for saw nn contacts exhibits a maximum near d 2 dimensions a behavior similar to previous observations for the saw exponent ensuremath gamma estimates of the ensuremath theta point for interacting saw s the critical temperature of the o m model and other lattice constants e g bond and site percolation thresholds are obtained in terms of saw and rw lattice parameters

a weak kantorovich existence theorem for the solution of nonlinear equations the kantorovich theorem is a fundamental tool in nonlinear analysis for proving the existence and uniqueness of solutions of nonlinear equations arising in various fields this theorem was weakened recently by argyros who used a combination of lipschitz and center lipschitz conditions in place of the lipschitz conditions of the kantorovich theorem in the present paper we prove a weak kantorovich type theorem that gives the same conclusions as the previous two results under weaker conditions illustrative examples are provided in the paper

extending the applicability of the local and semilocal convergence of newton s method we present a local as well a semilocal convergence analysis for newton s method in a banach space setting using the same lipschitz constants as in earlier studies we extend the applicability of newton s method as follows local case a larger radius is given as well as more precise error estimates on the distances involved semilocal case the convergence domain is extended the error estimates are tighter and the information on the location of the solution is at least as precise as before numerical examples further justify the theoretical results

self avoiding walk in five or more dimensions i the critical behaviour we use the lace expansion to study the standard self avoiding walk in thed dimensional hypercubic lattice ford 5 we prove that the numbercn ofn step self avoiding walks satisfiescn a n where is the connective constant i e 1 and that the mean square displacement is asymptotically linear in the number of steps i e v 1 2 a bound is obtained forcn x the number ofn step self avoiding walks ending atx the correlation length is shown to diverge asymptotically like z 1 2 the critical two point function is shown to decay at least as fast as x 2 and its fourier transform is shown to be asymptotic to a multiple ofk 2 ask 0 i e 0 we also prove that the scaling limit is gaussian in the sense of convergence in distribution to brownian motion the infinite self avoiding walk is constructed in this paper we prove these results assuming convergence of the lace expansion the convergence of the lace expansion is proved in a companion paper

local convergence of inexact newton methods a simple affine invariant condition which ensures local convergence of inexact newton methods is provided this condition is closely linked with the characterization of superlinear convergence given by dennis and more radius of convergence and rate of convergence results are derived and we show the theory is in some sense sharp we apply the theory to various inexact newton methods including difference newton like methods and the general newton like method in which the iterates are perturbed by rounding errors in the computation