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

metropolis monte carlo simulation of lattice animals we propose and study a metropolis monte carlo algorithm for the simulation of weighted lattice animals in in the canonical ensemble with a fixed number of edges we examine the implementation and performance of the algorithm data obtained by sampling uniformly weighted animals and animals weighted as critical percolation clusters are reported and analysed in particular we estimate autocorrelation times and dynamical exponents to study the efficiency of the algorithm and estimate the metric and branch exponents of lattice animals in two and three dimensions

the eigenvalue distribution of products of toeplitz matrices clustering and attraction abstract we use a recent result concerning the eigenvalues of a generic non hermitian complex perturbation of a bounded hermitian sequence of matrices to prove that the asymptotic spectrum of the product of toeplitz sequences whose symbols have a real valued essentially bounded product h is described by the function h in the szego way then using mergelyan s theorem we extend the result to the more general case where h belongs to the tilli class the same technique gives us the analogous result for sequences belonging to the algebra generated by toeplitz sequences if the symbols associated with the sequences are bounded and the global symbol h belongs to the tilli class a generalization to the case of multilevel matrix valued symbols and a study of the case of laurent polynomials not necessarily belonging to the tilli class are also given

characterizations of lie derivations of b x abstract let x be a banach space of dimension greater than 2 we prove that if b x b x is a linear map satisfying a b a b a b for any a b b x with ab 0 resp ab p where p is a fixed nontrivial idempotent then d where d is a derivation of b x and b x c i is a linear map vanishing at commutators a b with ab 0 resp ab p

characterizing homomorphisms derivations and multipliers in rings with idempotents in certain rings containing non central idempotents we characterize homomorphisms derivations and multipliers by their actions on elements satisfying some special conditions for example we consider the condition that an additive map h between rings mathcal a and mathcal b satisfies h x h y h z 0 whenever x y z in mathcal a are such that xy yz 0 as an application we obtain some new results on local derivations and local multipliers in particular we prove that if mathcal a is a prime ring containing a non trivial idempotent then every local derivation from mathcal a into itself is a derivation

commuting traces of biadditive mappings commutativity preserving mappings and lie mappings biadditive mappings b r r r where r is a prime ring with certain additional properties satisfying b x x x xb x x for all x r are characterized as an application we determine the structures of commutativity preserving mappings lie isomorphisms and lie derivations of certain prime rings

additive lie ξ lie derivations and generalized lie ξ lie derivations on nest algebras abstract for a scalar a notion of generalized lie derivations is introduced which coincides with the notion of generalized lie derivations if 1 some characterizations of additive generalized lie derivations on the triangular algebras and the standard operator subalgebras of banach space nest algebras are given it is shown under some suitable assumption that an additive map l is an additive generalized lie derivation if and only if it is the sum of an additive generalized derivation and an additive map from the algebra into its center vanishing all commutators is an additive generalized lie derivation with 1 if and only if it is an additive generalized derivation satisfying l a l a for all a

derivable maps and derivational points abstract for an algebra a and an a bimodule m let l a m be the set of all linear maps from a to m a map l a m is called derivable at c a if a b a b c for all a b a with ab c we call an element c a a derivational point of l a m if l a m the condition is derivable at c implies is a derivation we characterize derivable maps by means of peirce decompositions and determine derivational points for some general bimodules as a special case we see that for a nest algebra a on a hilbert space h every 0 c a is a derivational point of l a b h

monte carlo test of dimensional reduction for branched polymers in three dimensions by a monte carlo simulation of a grand canonical ensemble for a lattice tree model of branched polymers in three dimensions the author obtains the estimates theta 1 501 or 0 043 nu 0 495 or 0 013 for the critical exponents these estimates strongly support the exact conjecture theta 3 2 nu 1 2 by parisi and sourlas 1981 obtained using dimensional reduction the attrition constant mu is also estimated

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

enumerations of lattice animals and trees we have developed an improved algorithm that allows us to enumerate the number of site animals on the square lattice up to size 46 we also calculate the number of lattice trees up to size 44 and the radius of gyration of both lattice animals and trees up to size 42 analysis of the resulting series yields an improved estimate 4 062570 8 for the growth constant of lattice animals and 0 3 795254 8 for the growth constant of trees and confirms to a very high degree of certainty that both the animal and tree generating functions have a logarithmic divergence analysis of the radius of gyration series yields the estimate 0 64115 5 for the size exponent

the structure of lie derivations on c algebras we prove that every lie derivation on a c algebra is in standard form that is it can be

characterizations of lie derivations of triangular algebras abstract let t be a triangular algebra over a commutative ring r in this paper under some mild conditions on t we prove that if t t is an r linear map satisfying x y x y x y for any x y t with xy 0 resp xy p where p is the standard idempotent of t then d where d is a derivation of t and t z t where z t is the center of t is an r linear map vanishing at commutators x y with xy 0 resp xy p

the collapse transition for the lattice trees the authors consider a lattice tree model for the collapse of dilute branched polymers in the good solvent regime in which the collapse is driven by a near neighbour contact fugacity they describe some rigorous results including bounds for the temperature dependence of the reduced limiting free energy and compare these results with numerical estimates derived from exact enumeration data from the specific heat they estimate the collapse temperature tc and the cross over exponent phi 0 in two and three dimensions they find phi 0 0 60 or 0 03 d 2 and phi 0 0 82 or 0 03 d 3 finally they speculate on a possible roughening transition which may occur at a temperature tr tc

a rigorous bound on the critical exponent for the number of lattice trees animals and polygons the number ofn site lattice trees up to translation is believed to behave asymptotically ascn 0 n where is a critical exponent dependent only on the dimensiond of the lattice we present a rigorous proof that d 1 d for anyd 2 the method also applies to lattice animals site animals and two dimensional self avoiding polygons we also prove that v whend 2 wherev is the exponent for the radius of gyration

rooted spiral trees on hyper cubic lattices we study rooted spiral trees in 2 3 and 4 dimensions on a hyper cubic lattice using exact enumeration and monte carlo techniques on the square lattice we also obtain exact lower bound of 1 93565 on the growth constant series expansions give 1 3667 0 0010 and 0 6574 0 0010 on a square lattice with monte carlo simulations we get the estimates as 1 364 0 010 and 0 656 0 010 these results are numerical evidence against earlier proposed dimensional reduction by four in this problem in dimensions higher than two the spiral constraint can be implemented in two ways in either case our series expansion results do not support the proposed dimensional reduction

characterization of lie derivations on von neumann algebras abstract let m be any von neumann algebra without central summands of type i 1 for any scalar a characterization of any additive map l on m satisfies l ab ba l a b bl a al b l b a whenever ab 0 is given there exists an additive derivation such that 1 if 1 then l f where f is an additive map into the center vanishing on a b with ab 0 2 if 0 then l i z m and l a a l i a for all a 3 if is rational and 0 1 then l 4 if is not rational then i l i and l a a l i a

additive maps derivable at some points on j subspace lattice algebras abstract let l be a j subspace lattice on a real or complex banach space dim x with x 2 and alg l be the associated j subspace lattice algebra let alg l alg l be an additive map it is shown that if is derivable at zero point i e ab a b a b whenever ab 0 then a a a a where is an additive derivation and is a scalar if is generalized derivable at zero point i e ab a b a b a i b whenever ab 0 then is a generalized derivation it is also shown that if x is complex then every linear map derivable at unit operator on alg l is a derivation

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

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

commutativity up to a factor of bounded operators in complex hilbert space we explore commutativity up to a factor ab uba for bounded operators in a complex hilbert space conditions on the possible values of the factor u are formulated and shown to depend on spectral properties of the operators involved commutativity up to a unitary factor is considered for pairs of self adjoint operators examples of non trivial realizations of such commutation relations are given