tensor product theorems in positive characteristic abstract in this paper we study tensor products of t prime t ideals over infinite fields the behaviour of these tensor products over a field of characteristic 0 was described by kemer first we show using methods due to regev that such a description holds if one restricts oneself to multilinear polynomials only second applying graded polynomial identities we prove that the tensor product theorem fails for the t ideals of the algebras m 1 1 e and e e where e is the infinite dimensional grassmann algebra m 1 1 e consists of the 2 2 matrices over e having even i e central elements of e and the other diagonal consisting of odd anticommuting elements of e note that these proofs do not depend on the structure theory of t ideals but are elementary ones all this comes to show once more that the structure theory of t ideals is essentially about the multilinear polynomial identities

on the polynomial identities of the algebra m11 e abstract verbally prime algebras are important in pi theory they were described by kemer over a field k of characteristic zero 0 and k t the trivial ones m n k m n e m ab e here k t is the free associative algebra of infinite rank with free generators t e denotes the infinite dimensional grassmann algebra over k m n k and m n e are the n n matrices over k and over e respectively the algebras m ab e are subalgebras of m a b e see their definition below the generic also called relatively free algebras of these algebras have been studied extensively procesi described the generic algebra of m n k and lots of its properties models for the generic algebras of m n e and m ab e are also known but their structure remains quite unclear in this paper we study the generic algebra of m 11 e in two generators over a field of characteristic 0 in an earlier paper we proved that its centre is a direct sum of the field and a nilpotent ideal of the generic algebra and we gave a detailed description of this centre those results were obtained assuming the base field infinite and of characteristic different from 2 in this paper we study the polynomial identities satisfied by this generic algebra we exhibit a basis of its polynomial identities it turns out that this algebra is pi equivalent to a 5 dimensional algebra of certain upper triangular matrices the identities of the latter algebra have been studied these were described by gordienko as an application of our results we describe the subvarieties of the variety of unitary algebras generated by the generic algebra in two generators of m 11 e also we describe the polynomial identities in two variables of the algebra m 11 e

the centre of generic algebras of small pi algebras abstract verbally prime algebras are important in pi theory they are well known over a field k of characteristic zero 0 and k t the trivial ones m n k m n e m a b e here k t is the free associative algebra with free generators t e is the infinite dimensional grassmann algebra over k m n k and m n e are the n n matrices over k and over e respectively moreover m a b e are certain subalgebras of m a b e defined below the generic algebras of these algebras have been studied extensively procesi gave a very tight description of the generic algebra of m n k the situation is rather unclear for the remaining nontrivial verbally prime algebras in this paper we study the centre of the generic algebra of m 11 e in two generators we prove that this centre is a direct sum of the field and a nilpotent ideal of the generic algebra we describe the centre of this algebra as a corollary we obtain that this centre contains nonscalar elements thus we answer a question posed by berele

an approximate kam renormalization group scheme for hamiltonian systems we construct an approximate renormalization scheme for hamiltonian systems with two degrees of freedom this scheme is a combination of kolmogorov arnold moser kam theory and renormalization group techniques it makes the connection between the approximate renormalization procedure derived by escande and doveil and a systematic expansion of the transformation in particular we show that the two main approximations consisting in keeping only the quadratic terms in the actions and the two main resonances keep the essential information on the threshold of the breakup of invariant tori

elementary gradings on the lie algebra utn the algebras utn k utn k of the upper triangular matrices over a field k are of significant importance in the theory of algebras with polynomial identities group gradings on algebras appear in various areas and provide an indispensable tool in the study of the algebraic and combinatorial properties of the algebras in question in this paper we consider the lie algebra utn k utn k of all upper triangular matrices of order n we study the group gradings on this algebra it turns out that the gradings on the lie algebra utn k utn k are much more intricate than those in the associative case in this paper we describe the elementary gradings on the lie algebra utn k utn k finally we study the canonical grading on utn k utn k by the cyclic group znzn of order n we produce a finite basis of the graded polynomial identities satisfied by this grading

basis of the identities of the matrix algebra of order two over a field of characteristic p 2 abstract in this paper we prove that the polynomial identities of the matrix algebra of order 2 over an infinite field of characteristic p 2 admit a finite basis we exhibit a finite basis consisting of four identities and in almost all cases for p we describe a minimal basis consisting of two identities the only possibilities for p where we do not exhibit minimal bases of these identities are p 3 and p 5 we show that when p 3 one needs at least three identities and we conjecture a minimal basis in this case in the course of the proof we construct an explicit basis of the vector space of the central commutator polynomials modulo the ideal of the identities of the matrix algebra of order two

on the laplacian energy like invariant abstract let g be a connected graph of order n with laplacian eigenvalues 1 2 n 1 n 0 the laplacian energy like invariant of the graph g is defined as lel lel g i 1 n 1 i lower and upper bounds for lel are obtained in terms of n number of edges maximum vertex degree and number of spanning trees

determination of the threshold of the break up of invariant tori in a class of three frequency hamiltonian systems we consider a class of hamiltonians with three degrees of freedom that can be mapped into quasi periodically driven pendulums the purpose of this paper is to determine the threshold of the break up of invariant tori with a specific frequency vector we apply two techniques the frequency map analysis and renormalization group methods the renormalization transformation acting on a hamiltonian is a canonical change of coordinates which is a combination of a partial elimination of the irrelevant modes of the hamiltonian and a rescaling of phase space around the considered torus we give numerical evidence that the critical coupling at which the renormalization transformation starts to diverge is the same as the value given by the frequency map analysis for the break up of invariant tori furthermore we obtain by these methods numerical values of the threshold of the break up of the last invariant torus

algebras with involution with linear codimension growth we study the varieties of associative algebras with involution over a field of characteristic zero which are generated by a finite dimensional algebra in this setting we give a list of algebras classifying all such varieties whose sequence of codimensions is linearly bounded moreover we exhibit a finite list of algebras to be excluded from the varieties with such property as a consequence we find all possible linearly bounded codimension sequences

finitely generated algebras with involution and multiplicities bounded by a constant abstract let a be an algebra with involution over a field f of characteristic zero and let n a n 1 2 be the sequence of cocharacters of a for every n 1 let l n a denote the nth colength of a which is the sum of the multiplicities in n a in this article we classify in two different ways the finitely generated algebras satisfying an ordinary polynomial identity whose multiplicities of the cocharacters n a are bounded by a constant as a consequence this also yields a characterization of the varieties whose colength l n a n 1 2 is bounded by a constant

algebras with involution and multiplicities bounded by a constant abstract let a be an algebra with involution over a field of characteristic zero in this paper we characterize in two different ways when the multiplicities of the cocharacter of a are bounded by a constant as a consequence we characterize the algebras with involution of bounded colength

2 graded polynomial identities for the jordan algebra of the symmetric matrices of order two abstract the jordan algebra of the symmetric matrices of order two over a field k has two natural gradings by z 2 the cyclic group of order 2 we describe the graded polynomial identities for these two gradings when the base field is infinite and of characteristic different from 2 we exhibit bases for these identities in each of the two cases in one of the cases we perform a series of computations in order to reduce the problem to dealing with associators while in the other case one employs methods and results from invariant theory moreover we extend the latter grading to a z 2 grading on b n the jordan algebra of a symmetric bilinear form in a vector space of dimension n n 1 2 we call this grading the scalar one since its even part consists only of the scalars as a by product we obtain finite bases of the z 2 graded identities for b n in fact the last result describes the weak jordan polynomial identities for the pair b n v n

on the graded identities for elementary gradings in matrix algebras over infinite fields we find a basis for the g graded identities of the n n matrix algebra mn k over an infinite field k of characteristic p 0 with an elementary grading such that the neutral component corresponds to the diagonal of mn k

a star variety with almost polynomial growth abstract let f be a field of characteristic zero in this paper we construct a finite dimensional f algebra with involution m and we study its polynomial identities on one hand we determine a generator of the corresponding t ideal of the free algebra with involution and on the other we give a complete description of the multilinear identities through the representation theory of the hyperoctahedral group as an outcome of this study we show that the variety generated by m var m has almost polynomial growth i e the sequence of codimensions of m cannot be bounded by any polynomial function but any proper subvariety of var m has polynomial growth if g 2 is the algebra constructed in giambruno and mishchenko preprint we next prove that m and g 2 are the only two finite dimensional algebras with involution generating varieties with almost polynomial growth

on laplacian energy like invariant of a graph abstract let g be a simple graph of order n and let 1 2 n 0 be the laplacian spectrum of g the laplacian energy like invariant of g lel for short is defined as lel g i 1 n 1 i in this paper a new lower bound for lel of graphs in terms of the maximum degree is given meanwhile an upper bound and a lower bound for lel of the line graph resp the subdivision graph and the total graph of a regular graph g are obtained

central polynomials in the matrix algebra of order two abstract we exhibit minimal bases of the polynomial identities for the matrix algebra m 2 k of order two over an infinite field k of characteristic p 2 we show that when p 3 the t ideal of this algebra is generated by three independent identities and when p 3 one needs only two identities the standard identity of degree four and the hall identity note that the same holds when the base field is of characteristic 0 furthermore using the exact form of the basis of the identities for m 2 k we give finite minimal set of generators of the t space of the central polynomials for the algebra m 2 k the set of generators depends on the characteristic of the field as well

kolmogorov arnold moser renormalization group approach to the breakup of invariant tori in hamiltonian systems we analyze the breakup of invariant tori in hamiltonian systems with two degrees of freedom using a combination of kolmogorov arnold moser kam theory and renormalization group techniques we consider a class of hamiltonians quadratic in the action variables that is invariant under the chosen kam transformations following the approach of thirring the numerical implementation of the transformation shows that the kam iteration converges up to the critical coupling at which the torus breaks up by combining this iteration with a renormalization consisting of a shift of resonances and rescalings of momentum and energy we obtain a more efficient method that allows one to determine the critical coupling with high accuracy this transformation is based on the physical mechanism of the breakup of invariant tori we show that the critical surface of the transformation is the stable manifold of codimension one of a nontrivial fixed point and we discuss its universality properties

some results concerning the multiplicities of cocharacters of superalgebras with graded involution abstract let a be a finitely generated superalgebra with graded involution over a field f of characteristic zero and let n 1 n 4 a n 1 n 4 n be the n 1 n 4 cocharacter of a in this paper we present some results concerning the multiplicities of cocharacters for this kind of algebras

graded algebras with polynomial growth of their codimensions abstract let a be an algebra over a field of characteristic 0 and assume a is graded by a finite group g we study combinatorial and asymptotic properties of the g graded polynomial identities of a provided a is of polynomial growth of the sequence of its graded codimensions roughly speaking this means that the ideal of graded identities is very large we relate the polynomial growth of the codimensions to the module structure of the multilinear elements in the relatively free g graded algebra in the variety generated by a we describe the irreducible modules that can appear in the decomposition we show that their multiplicities are eventually constant depending on the shape obtained by the corresponding multipartition after removing its first row we relate moreover the polynomial growth to the colengths finally we describe in detail the algebras whose graded codimensions are of linear growth

minimal varieties and minimal supervarieties of polynomial growth abstract by a variety v we mean a supervariety or a variety generated by an associative algebra over a field f of characteristic zero in this case we consider its sequence of codimensions c n v and say that v is minimal of polynomial growth n k if c n v grows like n k but any proper subvariety grows like n t with t k in this paper we deal with minimal varieties generated by unitary algebras and prove that for k 2 there is only a finite number of them we also explicit a list of finite dimensional algebras generating such minimal varieties for k 3 we show that the number of minimal varieties can be infinity and we classify all minimal varieties of polynomial growth n k by giving a recipe for the construction of their t ideals

polynomial codimension growth of algebras with involutions and superinvolutions abstract let a be an associative algebra over a field f of characteristic zero endowed with a graded involution or a superinvolution and let c n a be its sequence of codimensions in 4 12 it was proved that if a is finite dimensional such sequence is polynomially bounded if and only if a generates a variety not containing a finite number of algebras the group algebra of z 2 and a 4 dimensional subalgebra of the 4 4 upper triangular matrices with suitable graded involutions or superinvolutions in this paper we focus our attention on such algebras since they are the only finite dimensional algebras up to t 2 equivalence generating varieties of almost polynomial growth i e varieties of exponential growth such that any proper subvariety has polynomial growth we classify the subvarieties of such varieties by giving a complete list of generating finite dimensional algebras along the way we classify all minimal varieties of polynomial growth and surprisingly we show that their number is finite for any given growth finally we describe the algebras whose codimensions are bounded by a linear function

semiclassical spectrum using secular perturbation theory the quartic oscillator we analyse the quartic oscillator using the technique of removal of resonances in a region of parameter space where the system exhibits soft chaos we obtain analytic expressions for all primary resonances and thus obtain a semiclassical spectrum a criterion relating to overlapping resonances gives an estimate about the break down of the removal of resonance method

on the laplacian coefficients of unicyclic graphs abstract let g be a graph of order n and let p g k 0 n 1 k c k n k be the characteristic polynomial of its laplacian matrix generalizing an approach of mohar on graph transformations we show that among all connected unicyclic graphs of order n the k th coefficient c k is largest when the graph is a cycle c n and smallest when the graph is the a s n with an additional edge between two of its pendent vertices a relation to the recently established laplacian like energy of a graph is discussed

generic 2 2 matrices in positive characteristic let r m k be the k algebra generated by the generic 2 2 matrices y 1 y m over a unitary commutative associative ring k our main result is that with a small class of exceptions for m a positive integer and p a prime the kernel of the natural homomorphism r m r m p coincides with pr m the only exceptions are for m 5 and p 2 when we give an explicit multilinear polynomial identity of degree 5 for the matrix algebra m 2 2 which does not follow from the spolynomial identities of m 2 2 this improves on schelter s construction of a non multilinear identity of this sort of degree 6 and drensky and tsiganchev s existence result for a multilinear identity such as we have found

a computational approach to construct a multivariate complete graph invariant in this paper we present a computational approach for finding complete graph invariants specifically we generate exhaustive sets of connected non isomorphic graphs with 9 and 10 vertices and demonstrate that a 97 dimensional multivariate graph invariant is capable to distinguish each of the non isomorphic graphs furthermore in order to tame the computational complexity of the problem caused by the vast number of graphs e g involving over 10 million networks with 10 vertices we suggest a low dimensional iterative procedure that is based on highly discriminative individual graph invariants we show that also this computational approach leads to a perfect discrimination overall our numerical results prove the existence of such graph invariants for networks with 9 and 10 vertices furthermore we show that our iterative approach has a polynomial time complexity

ordinary and graded cocharacter of the jordan algebra of 2 2 upper triangular matrices abstract let f be a field of characteristic zero and u j 2 f be the jordan algebra of 2 2 upper triangular matrices over f in this paper we give a complete description of the space of multilinear graded and ordinary identities in the language of young diagrams through the representation theory of a young subgroup of s n for every z 2 grading of u j 2 f we compute the multiplicities in the graded cocharacter sequence and furthermore we compute the ordinary cocharacter

a version of thirring s approach to the kolmogorov arnold moser theorem for quadratic hamiltonians with degenerate twist we give a proof of the kolmogorov arnold moser kam theorem on the existence of invariant tori for weakly perturbed hamiltonian systems based on thirring s approach for hamiltonians that are quadratic in the action variables the main point of this approach is that the iteration of canonical transformations on which the proof is based stays within the space of quadratic hamiltonians we show that thirring s proof for nondegenerate hamiltonians can be adapted to hamiltonians with degenerate twist this case in fact drastically simplifies thirring s proof

transformation properties of perturbation expansions around elliptic fixed points transformation properties of perturbation expansions of vibrational quasi periodic orbits upon the modular transformations of frequencies are studied using simple but nontrivial examples the siegel complex quadratic map and special solutions of the real area preserving quadratic map it is shown that the transformation properties are similar to those in the previously studied case of the rotational invariant tori except for some special features of the real map related to the atypical nature of 1 3 resonance in this case

graded identities of block triangular matrices abstract let f be an infinite field and u t d 1 d n be the algebra of upper block triangular matrices over f in this paper we describe a basis for the g graded polynomial identities of u t d 1 d n with an elementary grading induced by an n tuple of elements of a group g such that the neutral component corresponds to the diagonal of u t d 1 d n in particular we prove that the monomial identities of such algebra follow from the ones of degree up to 2 n 1 our results generalize for infinite fields of arbitrary characteristic previous results in the literature which were obtained for fields of characteristic zero and for particular g gradings in the characteristic zero case we also generalize results for the algebra u t d 1 d n c with a tensor product grading where c is a color commutative algebra generating the variety of all color commutative algebras

minimal algebras with respect to their exponent abstract given an m tuple a 1 a m of finite dimensional simple algebras we introduce a block triangular matrix algebra with involution denoted as ut a 1 a m where each a i can be embedded as algebra we describe the t ideal of r ut a 1 a m in terms of the ideals t a i and prove that any algebra with involution which is minimal with respect to its exponent is pi equivalent to r for a suitable choice of a 1 a m moreover we show that if m 1 or a i f for all i then r itself is a minimal algebra the assumption for the base field f is characteristic zero

quasiperiodic hamiltonian motions scale invariance harmonic oscillators the work of kolmogorov arnold and moser appeared just before the renormalization group approach to statistical mechanics was proposed by wilson phys rev b 4 3174 3183 1971 it can be classified as a multiscale approach which also appeared in works on the convergence of fourier s series l carleson acta math 116 135 157 1966 and c fefferman ann math 98 551 571 1973 or construction of euclidean quantum fields e nelson j math phys 5 1190 1964 or the scaling analysis of the short scale behavior of navier stokes fluids l caffarelli r kohn and l nirenberg commun pure appl math 35 771 831 1982 to name a few which resulted in numerous problems in this review the kam theorem proof will be presented as a classical renormalization problem with the harmonic oscillator as a trivial fixed point the work of kolmogorov arnold and moser appeared just before the renormalization group approach to statistical mechanics was proposed by wilson phys rev b 4 3174 3183 1971 it can be classified as a multiscale approach which also appeared in works on the convergence of fourier s series l carleson acta math 116 135 157 1966 and c fefferman ann math 98 551 571 1973 or construction of euclidean quantum fields e nelson j math phys 5 1190 1964 or the scaling analysis of the short scale behavior of navier stokes fluids l caffarelli r kohn and l nirenberg commun pure appl math 35 771 831 1982 to name a few which resulted in numerous problems in this review the kam theorem proof will be presented as a classical renormalization problem with the harmonic oscillator as a trivial fixed point