additive mappings on von neumann algebras preserving absolute values abstract given hilbert spaces h and k and a von neumann algebra a b h let denote the class of all additive mappings a b k satisfying a a a a the paper shows that if a contains no nonzero abelian central projection then every preserves the operation the r linear combination and up to a commuting operator multiple i 0 the ring multiplication if a contains a nonzero abelian central projection p and if the dimension of k is at least 2 or 2 rank p according to whether or not p can be chosen to be minimal then there exists an additive mapping a b k such that i is a projection and a a for all a a but is neither multiplicative nor adjoint preserving in case a b h the result was proved by molnar bull aust math soc 53 1996 391 when a contained all finite rank operators and by radjabalipour et al linear algebra appl 327 2001 197 under the redundant restriction i i k i k

lie triple derivations of tuhf algebras we study lie triple derivations of tuhf algebras it is shown that if l is a continuous lie triple derivation of a tuhf algebra t then there exists an associative derivation d of t such that l d where is a linear map of t into its center which annihilates brackets of operators

on maps preserving lie products equal to a rank one nilpotent abstract let be a bijective linear map on the algebra of n n complex matrices such that e 12 e 12 and a b e 12 whenever a b e 12 the purpose of this paper is to describe surprisingly has a different description from maps preserving zero lie products

nonlinear lie derivations of triangular algebras in this paper we prove that every nonlinear lie derivation of triangular algebras is the sum of an additive derivation and a map into its center sending commutators to zero

linear transformations on symmetric matrices that preserve commutativity it is shown that if a nonsingular linear transformation t on the space of n square real symmetric matrices preserves the commutativity where n 3 then t a qaqt q a in for all symmetric matricesa for some scalar orthogonal matrix q and linear functional q

jordan mappings of semiprime rings an additive mapping 8 of a ring r into a 2 torsion free ring r is called a jordan homomorphism if 6 ab z a 8 b 6 b a for all a be r a well known result of i n herstein 4 states that every jordan homomorphism onto a prime ring is either a homomorphism or an antihomomorphism suppose that r contains ideals u and v with null intersection let cp r u be a homomorphism and ii r v be an antihomomorphism a mapping 6 40 is a so called direct sum of mappings rp and 11 obviously b is a jordan homomorphism according to this construction we see that herstein s result 4 does not hold in semip me rings moreover w e baxter and w s martindale el showed by an example that a jordan homomorphism 0 of a ring r onto a semiprime ring r is not necessarily a direct sum of a homomorphism and an antihomomorphism but they proved that there always exists an essential ideal e of r such that the restriction of 8 to e is a direct sum of a homomorphism cp e r and an antihomomorphism y5 e r 1 theorem 2 71 in this paper theorem 2 3 we extend this result by showing that e can be choosen so that it is a sum of ideals u and v of r such that cp vanishes on v and tc vanishes on u even more for each x e r we have b ux e u e x for all u e u and b ux f x e o for all u e v this result also contains an affirmative answer to the question of baxter and martindale whether there is a way to choose the ideal e so that 8 e is an associative subring of r as we shall see in our case 6 e is the essential associative ideal of r an additive mapping 0 of a ring r into a ring r which satisfies b aba a 6 b 6 a for all a b e r will be called a jordan triple homomorphism an easy computation shows that every jordan homomorphism is also a

multiplicative lie n derivations of generalized matrix algebras abstract the aim of the paper is to give a description of multiplicative lie n derivations for a certain class of generalized matrix algebras as a consequence multiplicative lie n derivations of full matrix algebras are determined this solves a conjecture due to benkovic and eremita in 2012

functional identities in superalgebras abstract the aim of this paper is to establish the theory of functional identities in superalgebras as an application of this theory k supercommuting maps in superalgebras are investigated

generalized functional identities with anti automorphisms and derivations on prime rings i abstract the concept of a generalized functional identity gfi with anti automorphisms and derivations is a generalization of the notion of a generalized polynomial identity gpi with anti automorphisms and derivations in the present paper we show that either a prime ring is gpi or such gfis have only the obvious solutions our results simultaneously generalize the results of martindale rowen lanski kharchenko and chuang on gpis and a number of earlier results on gfis

generalized lie derivations on skew elements of prime algebras with involution we characterize generalized lie derivations on skew elements of prime algebras a with involution provided that a does not satisfy polynomial identities of low degree the analogous result for matrix algebras is also described

lie derivations on banach algebras abstract let d be a lie derivation on a unital complex banach algebra a then for every primitive ideal p of a except for a finite set of them which have finite codimension greater than one there exist a derivation d from a p to itself and a linear functional on a such that q p d a d a p a for all a a where q p denotes the quotient map from a onto a p moreover the preceding decomposition holds for all primitive ideals in the case where d is continuous

structure of weyl type lie algebras abstract let a be a unital commutative associative algebra over a field f of characteristic zero d a commutative subalgebra of der f a all derivations of the associative algebra a we assume that a is d simple and denote the center of the weyl type algebra a d by f 1 which is an extension field of f when a d is simple in this paper it is proved that the simple associative algebras a d are noncommutative domains and then the derivations of the simple associative algebras a d and of the associated lie algebras a d l are completely determined when dim f 1 f 1 d

lie triple derivations of nest algebras abstract let n be a nest on a complex separable hilbert space h and n be the associated nest algebra in this paper we prove that every lie triple derivation from n into itself is of the form x xt tx h x i where t n and h is a linear mapping from n into c such that h a b c 0 for all a b c n

a class of zero product determined lie algebras abstract let l be a lie algebra over a field f we say that l is zero product determined if for every f linear space v and every bilinear map l l v the following condition holds if x y 0 whenever x y 0 then there exists a linear map f from l l to v such that x y f x y for all x y l this article shows that every parabolic subalgebra p of a finite dimensional simple lie algebra defined over an algebraically closed field is always zero product determined applying this result we present a method different from that of wang et al 2010 9 to determine zero product derivations of p and we obtain a definitive solution for the problem of describing two sided commutativity preserving maps on p

commuting traces and commutativity preserving maps on triangular algebras abstract let a be a triangular algebra the problem of describing the form of a bilinear map b a a a satisfying b x x x x b x x for all x a is considered as an application commutativity preserving maps and lie isomorphisms of certain triangular algebras e g upper triangular matrix algebras and nest algebras are determined

strong commutativity preserving maps on lie ideals abstract let a be a prime ring and let r be a noncentral lie ideal of a an additive map f r a is called strong commutativity preserving scp on r if f x f y x y for all x y r in this paper we show that if f is scp on r then there exist c 2 1 and an additive map r z a such that f x x x for all x r where c is the extended centroid of a unless char a 2 and a satisfies the standard identity of degree 4

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

generalized lie derivations on triangular algebras let a be a unital algebra and let m be a unitary a bimodule we consider generalized lie derivations mapping from a to m our results are applied to triangular algebras in particular to nest algebras and block upper triangular matrix algebras we prove that under certain conditions each generalized lie derivation of a triangular algebra a is the sum of a generalized derivation and a central map which vanishes on all commutators of a

the occupation statistics for indistinguishable dumbbells on a rectangular lattice space i the method of hock and mcquistan used recently to solve the occupation statistics for indistinguishable dumbbells or dimers on a 2 2 n lattice is extended further to obtain for the l m n lattice general expressions for the normalization expectation and dispersion of the statistics and their limit as n becomes very large in particular an explicit expression of the partition function in the thermodynamic limit x is obtained for any value of the absolute activity x of dimers the developed mathematical formalism is then applied to planar lattices 1 m n with m 1 2 3 and 4 the known results for m 1 and 2 are recovered and some new ones are obtained the recurrence relation for the number a q n of arrangements of q dumbbells on a 1 m n lattice which has 3 and 5 terms when m 1 and 2 respectively is found to have 15 and 65 terms for m 3 and 4 analysis and extrapolation of the results enable one to predict the expectation 1mn on a planar 1 m n lattice to be 63 4 in the limit as both m a

idempotent elements determined matrix algebras abstract let m n r be the algebra of all n n matrices over a unital commutative ring r with 2 invertible v be an r module it is shown in this article that if a symmetric bilinear map from m n r m n r to v satisfies the condition that u u e u whenever u 2 u then there exists a linear map f from m n r to v such that x y f x y x y m n r applying the main result we prove that an invertible linear transformation on m n r preserves idempotent matrices if and only if it is a jordan automorphism and a linear transformation on m n r is a jordan derivation if and only if it is jordan derivable at all idempotent points

k commuting maps on triangular algebras abstract in this paper k commuting maps on certain triangular algebras are determined as an application we show that every k commuting map on an upper triangular matrix algebra over a unital commutative ring of 2 torsion free or a nest algebra is proper

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

on herstein s lie map conjectures i first published in transactions american mathematical society in vol 353 no 10 pp 4235 4260 published by the american mathematical society

nonlinear strong commutativity preserving maps on skew elements of prime rings with involution abstract let a be a prime ring of characteristic not 2 with involution with center z a and with skew elements k suppose that f k a is a map satisfying f x f y x y for all x y k then there exists a map k z a such that f x x x for all x k or f x x x for all x k except when a is an order in a 4 9 or 16 dimensional central simple algebra

on maps preserving zeros of the polynomial xy yx abstract let a m n f be the matrix algebra over a field f with an involution where n 20 suppose that a a is a bijective linear map such that x y y x for all x y a such that xy yx we show that is of the form x uxu 1 for x a where is a nonzero symmetric scalar and u is a normal matrix such that uu is a nonzero scalar

lie superhomomorphisms in superalgebras with superinvolution abstract the main purpose of the present paper is to give a description of lie superhomomorphisms from the set of skew elements of a superalgebra with superinvolution into a unital superalgebra

strong commutativity preserving maps in prime rings with involution abstract let a be a prime ring of characteristic not 2 with center z a and with involution let s be the set of symmetric elements of a suppose that f s a is an additive map such that f x f y x y for all x y s then unless a is an order in a 4 dimensional central simple algebra there exists an additive map s z a such that f x x x for all x s or f x x x for all x s

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

lie derivations of incidence algebras abstract let x be a locally finite preordered set r a commutative ring with identity and i x r the incidence algebra of x over r in this note we prove that each lie derivation of i x r is proper provided that r is 2 torsion free

nonlinear maps preserving lie products on factor von neumann algebras abstract in this paper we prove that every bijective map preserving lie products from a factor von neumann algebra m into another factor von neumann algebra n is of the form a a a where m n is an additive isomorphism or the negative of an additive anti isomorphism and m c i is a map with ab ba 0 for all a b m

linear maps preserving commutativity abstract commutativity preserving maps on the real space of all real symmetric or complex self adjoint matrices are characterized related results are given for adjoint preserving maps defined on all n n matrices these results are extended to infinite dimensions in the case of invertible maps

strong commutativity preserving maps on subsets of matrices that are not closed under addition abstract let m n d be the ring of all n n matrices over a division ring d where n 2 is an integer and let gl n d be the set of all invertible matrices in m n d we describe maps f gl n d m n d such that f x f y x y for all x y gl n d the analogous result for singular matrices is also obtained

on herstein s lie map conjectures iii abstract in the two papers on herstein s lie map conjectures i and ii solutions of all of herstein s problems concerning lie isomorphisms and derivations were given under the assumption that the algebras involved were of sufficiently high dimension in the present paper we remove this restriction thereby solving herstein s problems in full generality

commutativity preserving linear maps on central simple algebras abstract let f be a field with char f 0 and let a be a finite dimensional central simple algebra over f with dim f a 4 if a linear map a a satisfies x 2 x x x 2 for all x a in particular if preserves commutativity then is either a standard commutativity preserving map or its range is commutative

lie derivable maps on b x abstract let x be a banach space of dimension greater than 1 we prove that if a map b x b x satisfies a b a b a b for any a b b x then d where d is an additive derivation of b x and the map b x f i vanishes at commutators a b

an approximate solution of the monomer dimer problem on a square lattice ii the mathematical method developed in paper i is applied to obtain the partition function and thermodynamical properties of the monomer dimer problem for a square lattice in terms of the absolute activity x we also obtain by extrapolation an approximate expression of the partition function which is accurate to better than 0 1 in the range 0 x 10 the expectation of the statistics x is calculated in two different ways and numerical results agree to better than 3 thus showing the consistency of the underlying mathematical method consistent with earlier studies there is no phase transition approximation methods used in earlier work are also found to be in good agreement with our analytic study

maps on matrix spaces it is well known that every automorphism of the full matrix algebra is inner we give a short proof of this statement and discuss several extensions of this theorem including structural results for multiplicative maps on matrix algebras characterizations of monotone and orthogonality preserving maps on idempotent matrices some nonlinear preserver results and some recent theorems concerning geometry of matrices we show that all these topics are closely related and point out the connections with physics and geometry several open problems are posed

on lie admissible algebras whose commutator lie algebras are lie subalgebras of prime associative algebras abstract we describe third power associative multiplications on noncentral lie ideals of prime algebras and skew elements of prime algebras with involution provided that x y y x x y for all x y and the prime algebras in question do not satisfy polynomial identities of low degree we also obtain necessary and sufficient conditions for these multiplications to be fourth power associative or flexible

functional identities on tensor products of algebras abstract let r and s be unital algebras we show that if x is a d free subset of r and s is finite dimensional then the set x x s x x s s is a d free subset of the algebra r s the assumption that s is finite dimensional turns out to be necessary in general however we show that some important functional identities have only standard solutions on x even when s is infinite dimensional

herstein s lie theory revisited an associative ring r becomes a jordan ring r under x 0 y xy yx and a lie ring r under x y xy yx in case r has an involution i e an antiautomorphism of period 1 or 2 then the set of symmetric elements s s e r 1 s s is a jordan subring of r and the set of skew elements k ker 1 k k is a lie subring of rp in the early 1950s herstein initiated a study of the jordan and lie ideals of r s and k in case that r was a simple associative ring either without or with an involution in the ensuring years his work has been generalized in various directions on the one hand to the setting of prime and semiprime rings and on the other hand to invariance conditions other than that given by ideals besides herstein himself we mention lanski as having been a major force in this program their influence is certainly felt in the present paper we have taken the liberty of using various arguments without making specific mention from their papers notably 6 and s other important contributions in this area have also been made by baxter chacron erickson montgomery osborn streb and others part of our motivation in writing this paper is to obtain the lie ideal theory for semiprime rings with involution by a somewhat different approach from the self contained elementary very clever methods embodied in the original style of herstein let us describe very briefly the spirit of our approach for r a ring with involution and 1 any ideal i e an ideal invariant under in k k is always a lie ideal of k which we shall call a standard lie ideal of k a

the structure of nonlinear lie derivation on von neumann algebras national natural science foundation of china 11071201 youth national natural science foundation of china 11001230 youth talented natural science foundation of fujian 2008f3103 fundamental research funds for the central universities 2010121001

additive mappings on operator algebras preserving absolute values abstract it is shown that an additive map b h b k is the sum of two homomorphisms one of which is c linear and the other is c antilinear provided that 1 a a a for all a b h 2 b i is an orthogonal projection and 3 c i i k i k the structure of is more refined when it is injective the paper also studies the properties of in the absence of condition b here b h and b k denote the algebras of all bounded linear operators on hilbert spaces h and k respectively these extend a result of l molnar bull austral math soc 53 1996 391 saying an additive map b h b h is a constant multiple of an either c linear or c antilinear homomorphism provided that 1 a a a for all a b h and 2 b b h contains all finite rank operators

commutativity preserving operators on symmetric matrices abstract if is a nonsingular linear operator on n n symmetric matrices over a formally real field n 3 and if preserves commutativity then a cu 1 au f a i where u t u and c are scalar and is a linear functional this is an extension of a result of chan and lim

lie triple derivations of nest algebras on banach spaces abstract let n be a non trivial nest on x alg n be the associated nest algebra and l alg n b x be a linear mapping in this paper it is proved that l is a lie triple derivation if and only if there exist a derivation d alg n b x and a linear mapping h alg n c i with h x y z 0 for any x y z alg n such that l d h on alg n

characterizing additive ξ lie derivations of prime algebras by ξ lie zero products abstract let a be a unital prime algebra with a nontrivial idempotent over a field f for any scalar f all additive maps l a a satisfying l a b a l b 0 whenever a b 0 are characterized and the relation of l to the derivations are revealed where a b ab ba is the lie product of a b a

thermodynamics and molecular freedom of dimers on plane triangular lattices this is an extension of three previous papers dealing with dimers on rectangular lattices one two and three dimensions the technique presented in the first paper in this series continues to be fruitful for dimers on plane triangular lattices entropy isothermal compressibility constant pressure heat capacity and molecular freedom per dimer at close packing are obtained exactly for lattices infinite in one direction and finite in the other observations made in the third paper of the series concerning molecular freedom per dimer at close packing on rectangular lattices are used to extrapolate our results to infinite plane triangular lattices at close packing the molecular freedom per dimer on an infinite plane triangular lattice is calculated to be 2 356 527 in agreement with the value obtained by nagle based on our earlier findings the value of 2 356 527 was used to obtain the analytic fit for the thermodynamic quantities in terms of the normalized number density

on additive isomorphisms of prime rings preserving polynomials we study additive isomorphisms of prime rings preserving a multilinear polynomial of degree 2 our main theorem generalizes a number of results obtained for jordan n jordan lie and lie triple isomorphisms of prime rings

nonlinear lie derivations on factor von neumann algebras abstract in this paper we prove that every nonlinear lie derivation from a factor von neumann algebra into itself is an additive derivation

multiplicative lie n derivations of triangular rings abstract we introduce the notion of a multiplicative lie n derivation of a ring generalizing the notion of a lie triple derivation the main goal of the paper is to consider the question of when do all multiplicative lie n derivations of a triangular ring t have the so called standard form the main result is applied to the classical examples of triangular rings nest algebras and block upper triangular matrix rings

quasi identities on matrices and the cayley hamilton polynomial abstract we consider certain functional identities on the matrix algebra m n that are defined similarly as the trace identities except that the coefficients are arbitrary polynomials not necessarily those expressible by the traces the main issue is the question of whether such an identity is a consequence of the cayley hamilton identity we show that the answer is affirmative in several special cases and moreover for every such an identity p and every central polynomial c with zero constant term there exists m n such that the affirmative answer holds for c m p in general however the answer is negative we prove that there exist antisymmetric identities that do not follow from the cayley hamilton identity and give a complete description of a certain family of such identities

lie triple derivations of triangular algebras abstract let r be a commutative ring with identity a b be unital algebras over r and m be a unital a b bimodule which is faithful as a left a module and also as a right b module let t a m 0 b be the triangular algebra consisting of a b and m this work is motivated by some intensive works of bresar 4 cheung 9 and zhang et al 30 here we study lie triple derivations of t it is shown that under mild assumptions every lie triple derivation on t is of standard form that is l can be expressed through an additive derivation and a linear functional vanishing on all second commutators of t examples of lie triple derivations on some classical triangular algebras are supplied

on operators preserving commutativity abstract let l x be the algebra of all bounded operators on a non trivial complex banach space x and f l x l x a bijective linear operator such that f and f 1 both send commuting pairs of operators into commuting pairs then either f a uau 1 p a i or f a ua u 1 p a i where p is a linear functional on l x u is a bounded linear bijective operator between the appropriate two spaces is a complex constant and a is the adjoint of a the form of an operator f for which f and f 1 both send projections of rank one into projections of rank one is also determined

on generalized lie derivations of lie ideals of prime algebras abstract let a be a prime algebra of characteristic not 2 with extended centroid c let r be a noncentral lie ideal of a and let b be the subalgebra of a generated by r if f d r a are linear maps satisfying that f x y f x y f y x xd y yd x for all x y r then there exist a generalized derivation g b ac c and a linear map r c such that f x g x x for all x r and r r 0 provided that a does not satisfy the standard identity of degree 18

characterization of k commuting additive maps on rings abstract let k be a positive integer and r a ring having unit 1 denote by z r the center of r assume that the characteristic of r is not 2 and there is an idempotent element e r such that r satisfies a r e 0 a 0 a r 1 e 0 a 0 z e r e k z e r e and z 1 e r 1 e k z 1 e r 1 e then every additive map f r r is k commuting if and only if f x x h x for all x r where z r and h is an additive map from r into z r as applications all k commuting additive maps on prime rings and von neumann algebras are characterized

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

decoupling of a system of partial difference equations with constant coefficients and application consider d multi variable functions aj n j 1 to d where n stands for the evaluation point in the associated multi dimensional space of coordinates n1 n2 let us assume that the aj s satisfy a system of d linearly coupled finite difference equations the value of each function ai at the evaluation point n is given as a linear combination of the values of this function and others at shifted evaluation points by introducing d suitable generating functions gj j 1 to d one is able to replace the d coupled difference equations by a system of d linear equations where the gj s play the role of the d unknowns after solving this new system of equations it is then possible to construct a difference equation for each of the aj s relating the value of ai at the evaluation point n to the values of ai itself at shifted arguments the solution of such a decoupled equation can then be handled using the multi dimensional combinatorics function technique

lie ring isomorphisms between nest algebras on banach spaces let n and m be nests on banach spaces x and y over the real or complex field f and let algn and algm be the associated nest algebras respectively it is shown that a map algn algm is a lie ring isomorphism i e is additive lie multiplicative and bijective if and only if has the form a tat 1 h a i for all a algn or a ta t 1 h a i for all a algn where h is an additive functional vanishing on all commutators and t is an invertible bounded linear or conjugate linear operator when dimx t is a bijective linear transformation for some field automorphism of f when dimx

commutativity preserving maps we briefly describe several techniques that have been developed to solve the problem of characterizing commutativity preserving maps we prove a new result describing the general form of injective continuous commutativity preserving maps on full matrix algebras

short proofs of embeddings into exceptional groups of lie type the availability of effective character theoretic arguments makes the study of embeddings of a particular finite group into classical lie groups much easier than the study of its embeddings into exceptional lie groups in this paper our goal is to give short reasonably uniform character theoretic and module theoretic arguments to prove the existence of particular finite subgroups of exceptional groups of lie type as examples we apply the methods that we develop to give short proofs of the embeddings ru e7 5 hs e7 5 f4 2 e6 5 u3 8 e7 and j1 g2 11 apart from the much studied example j1 g2 11 these embeddings have previously only been settled on a case by case basis with intricate computation and group theoretic arguments our main tool is the following simple module theoretic condition which is sufficient to exhibit a g invariant subalgebra in a known g invariant lie algebra

commutativity preserving maps on lie ideals of prime algebras abstract one of the most studied linear preserver problems is the problem of characterizing maps preserving commutativity in this paper we study maps preserving commutativity on a lie ideal l of a prime algebra a we show by an example that in general we cannot expect the same standard conclusion as in the case when l a therefore we confine ourselves to some special classes of algebras where the usual result can be proved

multiplicative lie isomorphisms between factors abstract let m n be factors if m n is a multiplicative lie isomorphism then is of the form where is a linear isomorphism or a conjugate linear isomorphism or the negative of a linear anti isomorphism or the negative of a conjugate linear anti isomorphism of m onto n and is a mapping of m into c i i c which maps commutators into zero

structural ordering of interacting dimers on a square lattice we study the entropy of dimers on a semi infinite m n square lattice n to infinity with lattice interaction energy per dimer mu 1 and nearest neighbour interaction energy mu 2 numerical computations based on recursive transfer matrices show that for m 1 2 3 4 and 5 and under certain conditions the constant mu 2 curves of the entropy as a function of the coverage theta 1 and the fraction theta 2 of the maximum number of possible nearest neighbours merge into a single curve forming arches that meet at cusps at the highest points of these arches the interaction energies are in the ratio of small whole numbers based on the analysis of the configurations at the cusps we extrapolate the results to the infinite two dimensional lattice m infinity and obtain the location of at least two cusps the first cusp occurs at theta 1 1 2 theta 2 0 and zero entropy and the second at theta 1 2 3 theta 2 2 9 and an entropy of 0 102

semi centralizing maps of generalized matrix algebras abstract the present article is a continuation of our former work xiao and wei 2010 35 to some extent motivated by bresar s and cheung s wonderful ideas we will study semi centralizing maps of generalized matrix algebras and describe its general form by routine and complicated computations skew commuting maps and semi centralizing maps of generalized matrix algebras are specially considered we prove that any skew commuting map on a class of generalized matrix algebras is zero and that any semi centralizing derivation on a generalized matrix algebra is zero these results not only give new perspectives to the work of bresar 2004 6 but also extend the main results of cheung 2001 13 a number of applications related to semi centralizing maps are given

lie n derivations of unital algebras with idempotents abstract let a be a unital algebra with nontrivial idempotents we show that under certain assumptions every lie n derivation on a is of the form d where d is a derivation of a is both a singular jordan derivation and an antiderivation of a and is a linear map from a to its center z a that vanishes on a a a a as an application we give a description of lie n derivations of unital algebras with wide idempotents

on maps preserving zeros of lie polynomials of small degrees we describe maps preserving zeros of multilinear lie polynomials of degrees 3 and 4 on prime algebras and matrices over unital algebras in particular our theorems generalize several results related to commutativity preserving maps

commuting traces of multiadditive mappings abstract letrbe a prime ring andmann additive mapping onrsuch that m x x x 0 for allx r if char r 0 or n then there exist mappings iofrinto the extended centroidcofrsuch that m x x i 0 n i x x n i for allx r if in addition ris not algebraic of bounded degree n then for eachithere exists ani additive mappingmiofriintocsuch that i x mi x x for allx r

maps preserving zeros of a polynomial abstract let a be an algebra and let f x 1 x d be a multilinear polynomial in noncommuting indeterminates x i we consider the problem of describing linear maps a a that preserve zeros of f under certain technical restrictions we solve the problem for general polynomials f in the case where a m n f we also consider quite general algebras a but only for specific polynomials f

on functional identities in prime rings with involution abstract let a be a prime ring with involution let s be the symmetric elements let k be the skew elements let q ml be the maximal left ring of quotients x 1 x m noncommuting variables and e i f j g k h l a m 1 q ml i j k l 1 2 m we study functional identities of the form m i 1 e i i x i m j 1 x j f j j m k 1 g k k x k m l 1 x l h l l 0 for all x 1 x m a where e i i means e i x 1 x i x m etc in case s k is not algebraic of bounded degree 2 m definitive results are obtained as an application k commuting traces of symmetric n additive maps of either s or k into q ml are characterized

commuting centralizing traces and lie triple isomorphisms on triangular algebras revisited abstract the aim of the paper is to give a description of commuting centralizing traces of bilinear maps and lie triple isomorphisms on certain triangular algebras as consequences commuting centralizing traces of bilinear maps and lie triple isomorphisms on upper triangular matrix algebras over a 2 torsion free commutative ring are completely determined

functional identities in one variable abstract let a be a centrally closed prime algebra over a characteristic 0 field k and let q a a be the trace of a d linear map i e q x m x x where m a d a is a d linear map if q x x 0 for every x a then q is of the form q x i 0 d i x x i where each i is the trace of a d i linear map from a into k for infinite dimensional algebras and algebras of dimension d 2 this was proved by lee lin wang and wong in 1997 in this paper we cover the remaining case where the dimension is d 2 using this result we are able to handle general functional identities in one variable on a more specifically we describe the traces of d linear maps q i a a that satisfy i 0 m x i q i x x m i k for every x a

on bilinear maps on matrices with applications to commutativity preservers abstract let m n be the algebra of all n n matrices over a commutative unital ring c and let l be a c module various characterizations of bilinear maps m n m n l with the property that x y 0 whenever x any y commute are given as the main application of this result we obtain the definitive solution of the problem of describing not necessarily bijective commutativity preserving linear maps from m n into m n for the case where c is an arbitrary field moreover this description is valid in every finite dimensional central simple algebra