lie isomorphisms of nest algebras on banach spaces abstract we show that a lie isomorphism between nest algebras on banach spaces can be decomposed as where is an algebraic isomorphism or a negative of an algebraic anti isomorphism and is a linear map with image in the center vanishing on each commutator

nearly optimal solutions for the chow parameters problem and low weight approximation of halfspaces the chow parameters of a boolean function f 1 1 n 1 1 are its np1 degree 0 and degree 1 fourier coefficients it has been known since 1961 lchow 1961 tannenbaum 1961r that the exact values of the chow parameters of any linear threshold function f uniquely specify f within the space of all boolean functions but until recently lo donnell and servedio 2011r nothing was known about efficient algorithms for reconstructing f exactly or approximately from exact or approximate values of its chow parameters we refer to this reconstruction problem as the chow parameters problem our main result is a new algorithm for the chow parameters problem which given sufficiently accurate approximations to the chow parameters of any linear threshold function f runs in time o n2 1 e o log2 1 e and with high probability outputs a representation of an ltf f that is e close to f in hamming distance the only previous algorithm lo donnell and servedio 2011r had running time poly n 22o 1 e2 as a byproduct of our approach we show that for any linear threshold function f over 1 1 n there is a linear threshold function f which is e close to f and has all weights that are integers of magnitude at most n 1 e o log2 1 e this significantly improves the previous best result of diakonikolas and servedio l2009r which gave a poly n 2o 1 e2 3 weight bound and is close to the known lower bound of max n 1 e log log 1 e lgoldberg 2006 servedio 2007r our techniques also yield improved algorithms for related problems in learning theory in addition to being significantly stronger than previous work our results are obtained using conceptually simpler proofs the two main ingredients underlying our results are 1 a new structural result showing that for f any linear threshold function and g any bounded function if the chow parameters of f are close to the chow parameters of g then f is close to g 2 a new boosting like algorithm that given approximations to the chow parameters of a linear threshold function outputs a bounded function whose chow parameters are close to those of f

the eckmann hilton argument and higher operads abstract the classical eckmann hilton argument shows that two monoid structures on a set such that one is a homomorphism for the other coincide and moreover the resulting monoid is commutative this argument immediately gives a proof of the commutativity of the higher homotopy groups a reformulation of this argument in the language of higher categories is suppose we have a one object one arrow 2 category then its hom set is a commutative monoid a similar argument due to a joyal and r street shows that a one object one arrow tricategory is the same as a braided monoidal category in this paper we begin to investigate how one can extend this argument to arbitrary dimension we provide a simple categorical scheme which allows us to formalise the eckmann hilton type argument in terms of the calculation of left kan extensions in an appropriate 2 category then we apply this scheme to the case of n operads in the author s sense and classical symmetric operads we demonstrate that there exists a functor of symmetrisation sym n from a certain subcategory of n operads to the category of symmetric operads such that the category of one object one arrow one n 1 arrow algebras of a is isomorphic to the category of algebras of sym n a under some mild conditions we present an explicit formula for sym n a which involves taking the colimit over a remarkable categorical symmetric operad we will consider some applications of the methods developed to the theory of n fold loop spaces in the second paper of this series

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

monoidal globular categories as a natural environment for the theory of weakn categories we present a definition of weak categories based on a higher order generalization of apparatus of operads

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

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

symmetrisation of n operads and compactification of real configuration spaces abstract it is well known that the forgetful functor from symmetric operads to nonsymmetric operads has a left adjoint sym 1 given by product with the symmetric group operad it is also well known that this functor does not affect the category of algebras of the operad from the point of view of the author s theory of higher operads the nonsymmetric operads are 1 operads and sym 1 is the first term of the infinite series of left adjoint functors sym n called symmetrisation functors from n operads to symmetric operads with the property that the category of one object one arrow one n 1 arrow algebras of an n operad a is isomorphic to the category of algebras of sym n a in this paper we consider some geometrical and homotopical aspects of the symmetrisation of n operads we follow getzler and jones and consider their decomposition of the fulton macpherson operad of compactified real configuration spaces we construct an n operadic counterpart of this compactification which we call the getzler jones operad we study the properties of getzler jones operad and find that it is contractible and cofibrant in an appropriate model category the symmetrisation of the getzler jones operad turns out to be exactly the operad of fulton and macpherson these results should be considered as an extension of stasheff s theory of 1 fold loop spaces to n fold loop spaces n 2 we also show that a space x with an action of a contractible n operad has a natural structure of an algebra over an operad weakly equivalent to the little n disks operad a similar result holds for chain operads these results generalise the classical eckman hilton argument to arbitrary dimension finally we apply the techniques to the swiss cheese type operads introduced by voronov and prove analogous results in this case

inverse littlewood offord theorems and the condition number of random discrete matrices consider a random sum r v r nvn where 771 rin are independently and identically distributed i i d random signs and vi vn are integers the littlewood offord problem asks to maximize concentration probabilities such as p r i ih in this paper we develop an inverse littlewood offord theory somewhat in the spirit of freiman s inverse theory in additive combinatorics which starts with the hypothesis that a concentration probability is large and concludes that almost all of the v vn are efficiently contained in a generalized arithmetic progression as an application we give a new bound on the magnitude of the

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

iterated wreath product of the simplex category and iterated loop spaces generalising segal s approach to 1 fold loop spaces the homotopy theory of n fold loop spaces is shown to be equivalent to the homotopy theory of reduced tn spaces where tn is an iterated wreath product of the simplex category a sequence of functors from tn to g allows for an alternative description of the segal spectrum associated to a g space this yields a canonical reduced tn set model for each eilenberg maclane space

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

centers and homotopy centers in enriched monoidal categories abstract we consider a theory of centers and homotopy centers of monoids in monoidal categories which themselves are enriched in duoidal categories the duoidal categories introduced by aguiar and mahajan under the name 2 monoidal categories are categories with two monoidal structures which are related by some not necessary invertible coherence morphisms centers of monoids in this sense include many examples which are not classical in particular the 2 category of categories is an example of a center in our sense examples of homotopy center analogue of the classical hochschild complex include the gray category gray of 2 categories 2 functors and pseudonatural transformations and tamarkin s homotopy 2 category of d g categories d g functors and coherent d g transformations

lie isomorphisms in prime rings with involution abstract let r and r be prime rings with involutions of the first kind and with respective lie subrings of skew elements k and k furthermore assume rc c 1 4 9 16 25 64 where c is the extended centroid of r it is shown that any lie isomorphism of k onto k can be extended uniquely to an associative isomorphism of k onto k where k and k are respectively the associative subrings generated by k and k

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

linear preservers of zeros of matrix polynomials abstract we classify linear operators on matrices with semiring entries that preserve the zeros of multivariable matrix polynomials these matrix polynomials are defined via the adjoint operator x y xy yx as if the matrices were being considered as if over a field also we compare the results over fields with the results over semirings

iterated monoidal categories abstract we develop a notion of an n fold monoidal category and show that it corresponds in a precise way to the notion of an n fold loop space specifically the group completion of the nerve of such a category is an n fold loop space and free n fold monoidal categories give rise to a finite simplicial operad of the same homotopy type as the classical little cubes operad used to parametrize the higher h space structure of an n fold loop space we also show directly that this operad has the same homotopy type as the n th smith filtration of the barratt eccles operad and the n th filtration of berger s complete graph operad moreover this operad contains an equivalent preoperad which gives rise to milgram s small model for 2 2 x when n 2 and is very closely related to milgram s model of n n x for n 2

lie derivations of the skew elements of prime rings with involution abstract letrbe a prime ring with involution of the first kind and characteristic 2 3 letkdenote the skew elements ofr and letcdenote the extended centroid ofr assume that rc c 1 4 16 it is shown that any lie derivation ofkinto itself can be extended to an ordinary derivation of k counterexamples are given for the low dimension cases

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

bounded independence fools halfspaces we show that any distribution on 1 1 n that is k wise independent fools any halfspace or linear threshold function h 1 1 n to 1 1 i e any function of the form h x operatorname sign sum i 1 n w i x i theta where the w 1 dots w n and theta are arbitrary real numbers with error epsilon for k o epsilon 2 log 2 1 epsilon our result is tight up to log 1 epsilon factors using standard constructions of k wise independent distributions we obtain the first explicit pseudorandom generators g 1 1 s to 1 1 n that fool halfspaces specifically we fool halfspaces with error epsilon and seed length s k cdot log n o log n cdot epsilon 2 log 2 1 epsilon our approach combines classical tools from real approximation theory with structural results on halfspaces by servedio comput complexity 16 2007 pp 180 209

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

on generalized functional identities on prime rings abstract in the present paper we study generalized functional identities involving multi additive functions our results simultaneously generalize martindale s theorem on prime rings with generalized polynomial identities and bresar s results on generalized functional identities of degree two

maps preserving products xy yx on von neumann algebras nnsf of china 11071201 ynsf of china 11001230 fundamental research funds for the central universities 2010121001 ynsf of fujian 2008f3013

derivations modulo elementary operators abstract let r be a prime ring with extended centroid c and symmetric martindale quotient ring q s r suppose that q s r contains a nontrivial idempotent e such that e r r e r let r r r c c be the bi additive map x y g x y x h y i a i x b i y c i where g h r r are additive maps and where a i b i c i r c c are fixed suppose that is zero product preserving that is x y 0 for x y r with x y 0 then there exists a derivation r q s r c such that both g and h are equal to plus elementary operators moreover there is an additive map f r q s r c such that x y f x y for all x y r the result is a natural generalization of several related theorems in the literature actually we prove some more general theorems

vector analysis of threshold functions abstract viewing n variable boolean functions as vectors in r 2n we invoke basic tools from linear algebra and linear programming to derive new results on the realizability of boolean functions using threshold gates using this approach we obtain 1 a lower bound on the number of input functions required by a threshold gate implementing a given function 2 a lower bound on the error incurred when a boolean function is approximated by a linear combination of a set of functions 3 a limit on the effectiveness of a well known lower hound technique based on computing correlations among boolean functions for the depth of threshold circuits implementing boolean functions 4 a construction showing that every boolean function of n input variables is a threshold function of polynomially many input functions none of which is significantly correlated with 5 generalizations of some known results on threshold circuit complexity particularly those that are based on spectral analysis

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

operadic categories and duoidal deligne s conjecture abstract the purpose of this paper is two fold in part 1 we introduce a new theory of operadic categories and their operads this theory is in our opinion of an independent value in part 2 we use this new theory together with our previous results to prove that multiplicative 1 operads in duoidal categories admit under some mild conditions on the underlying monoidal category natural actions of contractible 2 operads the result of d tamarkin on the structure of dg categories as well as the classical deligne conjecture for the hochschild cohomology is a particular case of this statement

centralizing maps in prime rings with involution abstract let r be a prime ring with involution of characteristic 2 with center z skew elements k and extended centroid c theorem suppose k k 0 and k k is an additive map such that x x z for all x k then unless r is an order in a 16 dimensional central simple algebra there exists c and an additive map k c such that x x x for all x k

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

skew commutators and lie isomorphisms in real von neumann algebras in this paper we study skew elementsrkand their commutators in a real factorr and prove that ifrandsare real factors not of typei1andi2then derived lie algebras rk rk and sk sk are isomorphic if and only ifrandsare isomorphic

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

functional identities of degree 2 in arbitrary triangular rings abstract the aim of the paper is to give the solutions of functional identities of degree 2 in arbitrary triangular rings which generalizes a recent result due to eremita as applications commuting additive maps and generalized inner biderivations of arbitrary triangular rings are determined

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

the zelmanov approach to jordan homomorphisms of associative algebras abstract zelmanov s work on prime jordan algebras leads to an idempotent free version of martindale s theorem on the extension of jordan homomorphisms and derivations from the hermitian elements h r of an associative algebra of degree 2 with involution to associative homomorphisms and derivations on r the condition that j h r be of degree 2 is replaced by the intrinsic condition that j z j h a consist entirely of values of zelmanov polynomials

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 two sided infinite fixed points of morphisms let s be a finite alphabet and let h s s be a morphism finite and infinite fixed points of morphismsi e those words w such that h w wplay an important role in formal language theory head characterized the finite fixed points of h and later head and lando characterized the one sided infinite fixed points of h our paper has two main results first we complete the characterization of fixed points of morphisms by describing all two sided infinite fixed points of h for both the pointed and unpointed cases second we completely characterize the solutions to the equation h xy yx in finite words