geometry of free loci and factorization of noncommutative polynomials abstract the free singularity locus of a noncommutative polynomial f is defined to be the sequence of hypersurfaces z n f x m n k g det f x 0 the main theorem of this article shows that f is irreducible if and only if z n f is eventually irreducible a key step in the proof is an irreducibility result for linear pencils arising from this is a free singularity locus nullstellensatz for noncommutative polynomials apart from consequences to factorization in a free algebra the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry

on the graded identities and cocharacters of the algebra of 3 3 matrices abstract let m2 1 f be the algebra of 3 3 matrices over an algebraically closed field f of characteristic zero with non trivial z 2 grading we study the graded identities of this algebra through the representation theory of the hyperoctahedral group z 2 s n after splitting the space of multilinear polynomial identities into the sum of irreducibles under the z 2 s n action we determine all the irreducible z 2 s n characters appearing in this decomposition with non zero multiplicity we then apply this result in order to study the graded cocharacter of the grassmann envelope of m2 1 f finally using the representation theory of the general linear group we determine all the graded polynomial identities of the algebra m2 1 f up to degree 5

a restriction theorem for modules having a spherical submodule we introduce the following notion a finite dimensional represen tation v of a complex reductive algebraic group g is called spherical of rank one if the generic stabilizer m is reductive the pair g m is spherical and dim vm 1 let u be another finite dimensional representation of g we denote by s u s u g the ring of polynomial functions on u the ring of g invariant polynomial functions on u we characterize the image of s u v g under the restriction map into s u vm as the w ng m m invariants in the rees ring associated to an ascending filtration of s u m furthermore under some additional hypothesis we give an isomorphism be tween the graded ring associated to that filtration and s u p where p is the stabilizer of an unstable point whose 7 orbit has maximal dimension

identities and central polynomials with involution for the grassmann algebra abstract let e be the infinite dimensional grassmann algebra over an infinite field of characteristic p different from 2 given an involution on e denote by i d e and c e the set of all polynomial identities and central polynomials of e respectively in this paper we describe i d e and c e moreover we prove that c e is not finitely generated as a t space if p 2

excellent algebraic groups i abstract the notion of an excellent quadratic form has been introduced by m knebusch proc london math soc 33 1976 65 93 34 1977 1 31 it has an obvious analogue for a semisimple linear algebraic group g over an arbitrary field k the group g is excellent if for any field extension k of k the anisotropic kernel of the k group g k k can be defined by a k group the aim of this paper is to investigate excellence properties of special linear and orthogonal groups

applications of hook young diagrams to p i algebras abstract a the multiplicities m in the cocharacter n a of any p i algebra a are polynomially bounded b a hook containing n a b is obtained from the hooks containing n a and n b these results are obtained by applying a theory of hook young diagrams to p i algebras and they generalize results known for algebras satisfying capelli identities

matrix coefficient realization theory of noncommutative rational functions abstract noncommutative rational functions i e elements of the universal skew field of fractions of a free algebra can be defined through evaluations of noncommutative rational expressions on tuples of matrices this interpretation extends their traditionally important role in the theory of division rings and gives rise to their applications in other areas from free real algebraic geometry to systems and control theory if a noncommutative rational function is regular at the origin it can be described by a linear object called a realization in this article we present an extension of the realization theory that is applicable to arbitrary noncommutative rational functions and is well adapted for studying matrix evaluations of special interest are the minimal realizations which compensate the absence of a canonical form for noncommutative rational functions the non minimality of a realization is assessed by obstruction modules associated with it they enable us to devise an efficient method for obtaining minimal realizations with them we describe the stable extended domain of a noncommutative rational function and define a numerical invariant that measures its complexity using these results we determine concrete size bounds for rational identity testing construct minimal symmetric realizations and prove an effective local global principle for linear dependence of noncommutative rational functions

fuzzy interval finite element statistical energy analysis for mid frequency analysis of built up systems with mixed fuzzy and interval parameters abstract this paper introduces mixed fuzzy and interval parametric uncertainties into the fe components of the hybrid finite element statistical energy analysis fe sea model for mid frequency analysis of built up systems thus an uncertain ensemble combining non parametric with mixed fuzzy and interval parametric uncertainties comes into being a fuzzy interval finite element statistical energy analysis fife sea framework is proposed to obtain the uncertain responses of built up systems which are described as intervals with fuzzy bounds termed as fuzzy bounded intervals fbis in this paper based on the level cut technique a first order fuzzy interval perturbation fe sea ffipfe sea and a second order fuzzy interval perturbation fe sea method sfipfe sea are developed to handle the mixed parametric uncertainties efficiently ffipfe sea approximates the response functions by the first order taylor series while sfipfe sea improves the accuracy by considering the second order items of taylor series in which all the mixed second order items are neglected to further improve the accuracy a chebyshev fuzzy interval method cfim is proposed in which the chebyshev polynomials is used to approximate the response functions the fbis are eventually reconstructed by assembling the extrema solutions at all cut levels numerical results on two built up systems verify the effectiveness of the proposed methods

functional equations for character series associated with n n matrices l anneau a des invariants ou des traces des r matrices generiques n n possede une serie de caracteres a qui est une fonction rationnelle symetrique des variables commutantes x 1 x r si r n 2 cette serie de caracteres satisfait a une equation fonctionnelle

the henselization of a valued division algebra the main result of this paper is the following if f v is a valued field fh its henselization d a division algebra with center f and d f then v extends to d iff d ffh is a division algebra it is also shown that if p char f p 1if char f 0 then d f d f d f pb for some nonnegative integer b

finite element analysis with uncertain probabilities this paper presents a general method for the finite element analysis of linear mechanical systems by taking into account probability density functions whose parameters are affected by fuzziness within this framework the standard perturbation based stochastic finite element method is relaxed in order to incorporate uncertain probabilities in static dynamic and modal analyses general formulae are provided for assessing the fuzzy structural reliability and several typologies of optimization problems reliability based design robust design robust reliability based design are formalized in doing this the credibility theory is extensively used to extract qualified crisp data from the available set of fuzzy results so that standard optimizers can be adopted to solve the most important design problems it is shown that the proposed methodology is a general and versatile tool for finite element analyses because it is able to consider both probabilistic and non probabilistic sources of uncertainties such as randomness vagueness ambiguity and imprecision

the flat locus of brauer severi fibrations of smooth orders given a cayley hamilton smooth order a in a central simple algebra we determine the flat locus of the brauer severi fibration of a moreover we give a classification of all reduced central singularities where the flat locus differs from the azumaya locus and show that the fibers over the flat non azumaya points near these central singularities can be described as fibered products of graphs of projection maps this generalizes an old result of artin on the fibers of the brauer severi fibration of a maximal order over a ramified point finally we show these fibers are also toric quiver varieties and use this fact to compute their cohomology

polynomial identities for the jordan algebra of a symmetric bilinear form 1 the main purpose of this paper is to obtain a description of the relatively free algebras fj b and fj of the varieties of unitary jordan algebras 2b var g and 2 var gk the g module structure of these algebras is established and their hilbert or poincare series are computed a formula for the codimensions of 22 is given and the codimensions of i and 2bk are estimated asymptotically the varieties 2bk are described as subvarieties of 93 and lbk the proofs of these results use the representation theory of the symmetric and general linear groups the essential results are contained in sections 5 7 while the first four sections give the necessary preliminaries section 8 lists some open problems on varieties of jordan algebras motivated by results and problems from the associative theory 1

fuzzy based robust structural optimization robust optimization is conventionally defined as the collection of the possible problem solutions that can ensure acceptable performances and sufficient immunity against the effects of uncertain parameter variability methods proposed until now use a probabilistic way to model uncertainty and to quantify the final sensitivity in this work a fuzzy uncertainty modellization is adopted for structural engineering in particular to define solution performance scattering the fuzzy entropy is used as a global measure of variable dispersion the final formulation of the problem deals with two antithetical objective functions the fuzzy expected value of structural performance and its fuzzy entropy this fuzzy based approach in robust design is able to give a set of pareto optimal solutions in terms of structural efficiency and sensitivities regarding uncertainty and represents a suitable tool in supporting the decision maker finally different applications have been developed to demonstrate the applicability of the proposed method

on the codimension sequence of g simple algebras abstract let f be an algebraically closed field of characteristic zero and let g be a finite group in this paper we will show that the asymptotics of c n g a the g graded codimension sequence of a finite dimensional g simple f algebra a has the form n 1 dim f a e 2 dim f a n as conjectured by e aljadeff d haile and m natapov where is some positive real number and a e denotes the identity component of a in the special case where a is the algebra of m m matrices with an arbitrary elementary g grading we succeeded in calculating the constant explicitly

footnotes to a paper of domokos i abstract we use regev s double centralizer theorem from 7 to study invariants of matrices with group actions this theory then has applications to trace identities and poincare series

a random fuzzy analysis of existing structures abstract two approaches are proposed to estimate the reliability of existing structures by considering both the randomness in some of the design parameters and the fuzzy imprecision in some other parameters representing the in place condition of the aged structures in the first approach the fuzzy imprecision is transformed into random uncertainty using the entropy concept and the reliability of existing structures is estimated using well established theories of probability in the second approach a hybrid approach in the random fuzzy domain is used to evaluate the reliability using an level concept the multiple fuzzy variables case is also considered both discrete and continuous fuzzy variables are considered the results obtained from the proposed approaches are compared with other techniques available in the literature whenever possible both methods are applied to civil structural engineering problems in this paper the results obtained are very encouraging and demonstrate the applicability and robustness of the algorithms

expected value of fuzzy variable and fuzzy expected value models this paper will present a novel concept of expected values of fuzzy variables which is essentially a type of choquet integral and coincides with that of random variables in order to calculate the expected value of general fuzzy variable a fuzzy simulation technique is also designed finally we construct a spectrum of fuzzy expected value models and integrate fuzzy simulation neural network and genetic algorithms to produce a hybrid intelligent algorithm for solving general fuzzy expected value models

multiplicities of schur functions in invariants of two 3 3 matrices abstract we relate with any symmetric function f x y c x y presented as an infinite linear combination of schur functions f x y m 1 2 s 1 2 x y the multiplicity series m f m 1 2 t 1u 2 we study the behavior of m f under natural combinatorial and algebraic constructions in particular we calculate the multiplicity series for the symmetric algebra of the irreducible gl 2 c module corresponding to the complete symmetric function of degree 3 our main result is that we have found the explicit form of the multiplicity series for the hilbert or poincare series of the algebra of invariants of two 3 3 matrices as a consequence we have precised the result of berele on the asymptotics of the multiplicities in the trace cocharacter sequence of two 3 3 matrices

invariant theory for matrices over the grassmann algebra we prove that the algebra of polynomial maps mn e e invariant under simultaneous conjugation is generated by the queer trace polynomials

an efficient approach for design optimization of structures involving fuzzy variables abstract we propose a new approach for solving design optimization problems of structures involving fuzzy variables first the possibilistic safety model of structures with fuzzy variables is investigated a possibilistic safety index psi is presented to compare different designs this paper then presents the psi based design optimization psibdo model for structures with fuzzy variables the optimization problem is a triple loop nested problem and the computational cost is high to reduce the computation time a technique called target performance based design approach tpbda is proposed to overcome difficulties encountered by the psibdo with the proposed tpbda the triple loop nested problem is reduced to a double loop one and the computational time is considerably reduced several examples are given to demonstrate the efficiency of the proposed approach

poincare series of some pure and mixed trace algebras of two generic matrices we work over a field k of characteristic zero the poincare series for the algebra cn 2 of gln invariants and the algebra tn 2 of gln concomitants of two generic n n matrices x and y are computed for n 6 both simply graded and bigraded cases are included the cases n 4 were known previously for c4 2 and c5 2 we construct a minimal set of generators and give an application to specht s theorem on unitary similarity of matrices by identifying the space mn2 of pairs of n n matrices with mn k2 we extend the action of gln to gln gl2 for n 5 we compute the poincare series for the polynomial invariants of this action when restricted to the subgroups gln sl2 and gln 1 where 1 is the maximal torus of sl2 consisting of diagonal matrices five conjectures are proposed concerning the numerators and denominators of various poincare series mentioned above some heuristic formulas and open problems are stated

division algebras over henselian fields in this chapter we focus on the tame division algebras d with center a field f with henselian valuation v as usual we approach this by first obtaining results for graded division algebras then lifting back from operatorname mathsf gr d to d this is facilitated by results in 8 1 on existence and uniqueness of lifts of tame subalgebras from operatorname mathsf gr d to d in 8 2 we describe four fundamental canonical up to conjugacy subalgebras of d that reflect its valuative structure the rest of the chapter is devoted to brauer group factorizations of d corresponding to the noncanonical direct product decomposition of operatorname mathit br t f given in cor 7 85 the factor operatorname mathit hom c operatorname mathcal g overline f mathbb t gamma f is represented by a type of division algebra n called decomposably semiramified defined in 8 3 and characterized by the property that n contains a maximal subfield inertial over f and another totally ramified over f we show in 8 4 that every tame division algebra d is brauer equivalent to some s f t where s is inertially split and t is tame and totally ramified over f we show further that every inertially split division algebra s is brauer equivalent to some i f n where i is inertial over f and n is decomposably semiramified the classes overline i for the i appearing in the i f n decompositions of s are shown to range over a single coset of mathit dec z overline s overline f in operatorname mathit br overline f called the specialization coset of s in the final subsection 8 4 6 we summarize what happens in the special case that v is discrete of rank 1 where substantial simplifications occur

cohen macaulayness of modules of invariants for sl2 abstract let u w be finite dimensional representations of g sl2 we give conditions under which u k w g is a cohen macaulay k w g module in particular we obtain an invariant theoretic proof of the fact that the trace ring of generic 2 2 matrices is cohen macaulay

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

asymptotic codimensions of mk e abstract we show that the codimension sequence of the algebra of k k matrices over the grassmann algebra c n m k e is asymptotic to n 1 k 2 2 2 k 2 n where is an undetermined constant

a description of the brauer severi scheme of trace rings of generic matrices abstract here we use invariant theory to describe the brauer severi scheme of the fibers of trace rings of generic matrices with an algebraically closed base field of characteristic zero when the trace ring is viewed as a sheaf of algebras over the variety of matrix invariants using this approach we first prove that the brauer severi scheme of a trace ring is isomorphic to proj q for a graded ring q whose generators we describe in the first section this description also has a relevant interpretation over base fields of arbitrary characteristic in the second section of this paper we show that the brauer severi scheme of the fiber of a trace ring over a point that is not too degenerate will have smooth irreducible components meeting transversally and describe these irreducible components as brauer severi schemes of certain algebras

the ring of generic matrices in chapter 3 we defined a single n x n generic matrix we now introduce the ring generated by a set of n x n generic matrices

strongly nonlinear vortices in the stokes layer on an oscillating cylinder the stokes layer on a torsionally oscillating circular cylinder is susceptible to axially periodic vortices which are driven by a centrifugal mechanism which has some similarities to that which is responsible for gortler vortices in boundary layers on concave surfaces for small wavelength vortices an analytic description of linear weakly nonlinear and fully nonlinear perturbations is possible and as the growth rates of these modes are asymptotically greater than the timescale of the underlying stokes layer a quasi steady stability analysis is attempted in preference to a floquet type approach it is shown that highly nonlinear vortices can be expected to form on the surface of the cylinder it is noted that these vortices alter the basic mean flow at leading order and occupy a zone which has depth comparable to the stokes thickness depending on the taylor number of the flow three distinct types of vortex behaviour are found in the first of these corresponding to taylor numbers just above that at which the stokes layer becomes unstable over a part of its cycle the vortices are relatively weak and remain attached to the cylinder throughout their lifetime at larger taylor numbers the vortices initially grow whilst remaining on the cylinder but at some stage they break away and concentrate in a region within the stokes layer however their strength reduces with time and they soon decay away in the third regime identified before this vortex activity region dies the residual flow next to the cylinder becomes unstable once more in this case a second vortex zone begins to grow so that for a certain time interval two active zones exist the governing equations for the strongly nonlinear modes take the form of a free boundary value problem subject to ordinary differential constraints numerical solutions are presented and although periodic solutions are by no means guaranteed by use of the quasi steady approach the computations show that remarkably periodic vortex configurations tend to result

valuation rings in finite dimensional division algebras a subring b of a division algebra d is called a valuation ring of d if x or x 1 is contained in b for every x in d x 0 such a ring is called an extension of the valuation ring b of k the centre of d if b k b let d be a division algebra finite dimensional over its centre k d k n2 b a valuation ring of k and b bi iei the set of all extension of b in d theorem 1 b possesses at most n extensions in d i e b n theorem 2 any two extensions of b in d are conjugate in d theorems 3 and 4 the set t of elements in d which are integral over b is a subring of d if and only if b 1 in this case t bi bi in b

unramified cohomology of quadrics i given a quadric x over a field f of characteristic 2 we compute the kernel and cokernel of the natural map in degree 4 from the mod 2 galois cohomology of f to the unramified mod 2 cohomology of f x when dim x 10 and in several smaller dimensional cases applications of these results to real quadrics and to the unramified witt ring are given

trace cocharacters and the kronecker products of schur functions abstract it follows from the theory of trace identities developed by procesi and razmyslov that the trace cocharacters arising from the trace identities of the algebra mr f of r r matrices over a field f of characteristic zero are given by tcr n r n where denotes the kronecker product of the irreducible characters of the symmetric group associated with the partition with itself and r n denotes the set of partitions of n with r or fewer parts i e the set of partitions 1 k with k r we study the behavior of the sequence of trace cocharacters tcr n in particular we study the behavior of the coefficient of n m in tcr n as a function of n where 1 k is some fixed partition of m and n m k our main result shows that such coefficients always grow as a polynomial in n of degree r 1

robust design of structures using optimization methods the robust design of structures with stochastic parameters is studied using optimization techniques the first two statistical moments of the stochastic parameters including design variables are considered in conjunction with the second order perturbation method for the approximation of mean value and variance of the structural response in this framework the sensitivities of the mean values and variances of the structural performance function with respect to the design variables are calculated for use in the optimization task the robust design of structures is formulated as a multi criteria optimization problem in which both the expected value and the standard deviation of the objective function are to be minimized the robustness of the feasibility is also taken into account by involving the variability of the structural response in the constraints the two criteria optimization problem is converted into a scalar one and is then solved by a gradient based optimization algorithm to demonstrate the applicability of the presented method numerical examples are given involving static and dynamic response

colength sequences for matrices 1 a berele homogeneous polynomial identities israel j math 42 1982 258 272 2 a berele a regev some remarks on trace cocharacters j algebra 176 1995 1013 1024 3 a berele d j saltman the centers of generic division algebras with involution israel j math 63 1988 98 118 4 v s drensky codimensions of t ideals and hilbert series of relatively free algebras j algebra 91 1984 1 17 5 e formanek invariants and the ring of generic matrices j algebra 89 1984 178 223 6 e formanek a conjecture of regev about the capelli polynomial j algebra 109 1987 93 114 7 i n herstein notes from a ring theory conference cbms vol 9 amer math soc 1971 8 g james a kerber the representation theory of the symmetric group addison wesley reading ma 1981 9 i g macdonald symmetric functions and hall polynomials second ed oxford university press claren don 1995

maximal multiplicities in cocharacter sequences if a is a p i algebra mn a is the maximum multiplicity of the sn irreducible characters in the nth cocharacter of a we prove that if 1 a then mn a is asymptotic to a polynomial and we compute the degree of this polynomial for the verbally prime p i algebras

minimal varieties of algebras of exponential growth abstract the exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer through this scale we can now classify all minimal varieties of given exponent and of finite basic rank as a consequence we describe the corresponding t ideals of the free algebra and we compute the asymptotics of the related codimension sequences verifying in this setting some known conjectures we also show that the number of these minimal varieties is finite for any given exponent we finally point out some relations between the exponent of a variety and the gelfand kirillov dimension of the corresponding relatively free algebras of finite rank

a conjecture of regev about the capelli polynomial abstract let x1 xn2 y1 yn2 be generic n n matrices over a field k of characteristic zero if x 1 x n is a multilinear invariant of x1 xn then s z 2 sign f x 1 y 1 x 2 y 2 y 3 y 4 x 3 y 5 y 9 x n y n 2 2n 2 y n 2 f x 1 x n y where sn2 is the symmetric group of degree n2 y is the discriminant of y1 yn2 and tf x 1 x n is a uniquely defined multilinear invariant of x1 xn thus tf defines a function from the vector space of multilinear invariants of x1 xn to itself an analysis of this function is used to prove regev s conjecture that s z 2 sign x n 1 y 1 x 2 x 3 x 4 y 2 y 3 y 4 x 5 x n 2 2 2 x n 2 y n 2 2n 2 y n 2 is nonzero in addition a variant of the above function is used to evaluate the capelli polynomial

large wavenumber vortices in time periodic flows the linear stability properties of short wavelength vortices within time periodic flows are examined using a wkb based asymptotic analysis such vortices typically have an instantaneous growth rate considerably larger than the time scale of the underlying flow and thus have often been studied using quasi steady techniques a major drawback of these quasi steady approaches is that although they are able to determine the instantaneous flow structure they are of no use in describing the evolution of the modes it is shown how rational asymptotic analysis based on wkb ideas overcomes this deficiency and enables the form of the neutral stability curve to be obtained for short wavelengths the technique is applied to three problems first the key features of the analysis are exemplified by careful consideration of a model problem which is motivated by the physical situation of the sinusoidal heating of a horizontal flat plate bounding a semi infinite layer of fluid the governing linearized stability equations take the form of a system which is sixth order in the spatial variable and second order in time however the important features of this system are captured by our model which is only second order in the spatial coordinate and first order in time the second physically important flow examined is the motion induced in a viscous fluid surrounding a long torsionally oscillating cylinder a curved stokes flow detailed asymptotic and numerical studies of both the model and the two physically motivated flows are undertaken and it is shown how the analytical results provide good qualitative agreement with the numerical findings at quite modest wavenumbers the methodology adopted here may be used as a basis for the rational study of the stability properties of other time periodic flows a key result forthcoming from this study is the demonstration that the first few terms in the relevant high wavenumber form of the neutral curve for modes in an oscillatory flow may be derived relatively quickly with minimal computational effort

regularity of trace rings of generic matrices is called the ring of m generic n x n matrices g n the f subalgebra of m em generated by g and tr g is the ring of m generic n x n matrices and is denoted by t n these trace rings appear naturally in the study of finite dimensional representations of free algebras and in the invariant theory of n x n matrices 7 unlike rings of generic matrices t n shares some properties with commutative polynomial rings e g they are maximal orders and even unique factorization rings in the sense of chatters and jordan however their homological properties are far from being understood the main aim of this paper is to prove the following result

defining relations of invariants of two 3 3 matrices over a field of characteristic 0 teranishi 1986 found a minimal system of eleven generators of the algebra of invariants of two 3 3 matrices under simultaneous conjugation by gl3 nakamoto 2002 obtained the explicit but very complicated defining relation for a similar system of generators over z in this paper we give another natural set of eleven generators of the algebra of invariants over a field of characteristic 0 and the defining relation with respect to this generating set our defining relation is much simpler than that of nakamoto

extensions of valuation rings in central simple algebras certain subrings r of simple algebras q finite dimensional over their center k are studied these rings are called q valuation rings since they share many properties with commutative valuation rings let v be a valuation ring of k the center of q and let m be the set of q valuation rings r in q with r n k v then 1s 1 this extension theorem which does not hold if one considers only total valuation rings was proved by n i dubrovin here first a somewhat different proof of this result is given and then information about the set m is obtained theorem the elements in j are conjugate if v has finite rank theorem the elements in m are total valuation rings if m contains one total valuation ring in this case q is a division ring theorem i i 1 if 5 contains an invariant total valuation ring

uncertain buckling its past present and future an authoritative review on stochastic buckling of structures was written by amazigo some quarter century ago the present review summarizes some of the developments which took place since then a brief overview of the effect of uncertainty in the initial geometric imperfections elastic moduli applied forces and thickness variation is given for the benefit of the thoughtful reader the review is of critical nature

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

remarks on p i algebras over finite fields let a be a p i algebra over a finite field f let f r rp be the residue field of a discrete valuation ring d v r r and let f r r1 r2 where r1 r2 are d v r s assume a bpb for some r p i algebra b we form an rn a and bn rn b then study the limits for the identities and codimensions in the corresponding spaces thus we relate p i algebras over the rationals q to p i algebras over finite fields

fuzzy dynamical system approach for a dual parameter hybrid order robust control design abstract robust control for uncertain dynamical systems is considered the uncertainty is assumed to be bounded and the fuzzy set theory is adopted to describe the bound a new dual parameter hybrid order robust control design is then proposed in two steps first a class of robust controls is designed taking the advantage of the flexibility in choosing the order and a design parameter the controls are deterministic and are not if then fuzzy rules based deterministic performance including uniform boundedness and uniform ultimate boundedness for the controlled systems is guaranteed second a fuzzy set theoretic based performance index consisting of average and steady state system performance and control cost is proposed a constrained optimization problem is formulated with the intension of minimizing the performance index the objective of the optimization problem is to search for the optimal choice of dual parameters that minimizes the performance index we show that the solution to the optimization problem always exists as a result the resulting control design achieves two concurrent goals guaranteeing deterministic performance as well as minimizing the performance index the theory is illustrated by an unmanned aerial vehicle uav system control design

poincare series of semi invariants of 2 2 matrices abstract the poincare series of the algebra of sl 2 c sl 2 c invariants of m tuples of 2 2 matrices is presented both as a rational function and as a series of schur functions we show that this algebra of invariants is generated by the determinants the mixed discriminants and the discriminants of 2 2 matrices consequences on invariants of three dimensional matrices of the shape 2 2 m are discussed for arbitrary n 2 we prove an explicit functional equation for the poincare series of the sl n c sl n c invariants of m tuples of n n matrices

stable rationality of certain pgln quotients stable rationality of the field of matrix invariants m n m n pgl n is proved forn 5 andn 7 in combination with existing results this shows that v pgl n is stably rational wheneverv is an almost free representation ofpgl n andn divides 420 22 3 5 7

non commutative resolutions of quotient singularities for reductive groups in this paper we generalize standard results about non commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups we show in particular that quotient singularities for reductive groups always have non commutative resolutions in an appropriate sense moreover we exhibit a large class of such singularities which have twisted non commutative crepant resolutions we discuss a number of examples both new and old that can be treated using our methods notably we prove that twisted non commutative crepant resolutions exist in previously unknown cases for determinantal varieties of symmetric and skew symmetric matrices in contrast to almost all prior results in this area our techniques are algebraic and do not depend on knowing a commutative resolution of the singularity

the structure of highly nonlinear vortices in curved channel flow the structure of highly nonlinear small wavelength gortler vortices is analysed for fully developed flow in a curved channel the taylor number is taken to be t 0 epsilon 5 which is asymptotically larger than the neutral stability taylor number for vortices of small wavelength 2 pi epsilon over most of the flow domain the motion is determined by the interaction of the mean flow and terms proportional to exp iz epsilon whereas adjacent to the outer channel wall there is a boundary layer of thickness o epsilon which requires all the harmonics of the disturbance to be considered numerical solutions of the wall layer equations are presented for t 0 up to 14 000 with the main feature of these results being the development of a weak vortex on the boundary layer lengthscale

birational classification of fields of invariants for groups of order 128 let g be a finite group acting on the rational function field c xg g g c xg g g by cc automorphisms h xg xhgh xg xhg for any g h gg h g noether s problem asks whether the invariant field c g k xg g g gc g k xg g g g is rational i e purely transcendental over cc by fischer s theorem c g c g is rational over cc when g is a finite abelian group saltman and bogomolov respectively showed that for any prime p there exist groups g of order p9p9 and of order p6p6 such that c g c g is not rational over cc by showing the non vanishing of the unramified brauer group brnr c g 0brnr c g 0 which is an avatar of the birational invariant h3 x z torsh3 x z tors given by artin and mumford where x is a smooth projective complex variety whose function field is c g c g for p 2p 2 chu hu kang and prokhorov proved that if g is a 2 group of order 32 then c g c g is rational over cc chu hu kang and kunyavskii showed that if g is of order 64 then c g c g is rational over cc except for the groups g belonging to the two isoclinism families 13 13 with brnr c g 0brnr c g 0 and 16 16 with brnr c g c2brnr c g c2 bogomolov and bohning s theorem claims that if g1g1 and g2g2 belong to the same isoclinism family then c g1 c g1 and c g2 c g2 are stably cc isomorphic we investigate the birational classification of c g c g for groups g of order 128 with brnr c g 0brnr c g 0 moravec showed that there exist exactly 220 groups g of order 128 with brnr c g 0brnr c g 0 forming 11 isoclinism families j j we show that if g1g1 and g2g2 belong to 16 16 31 31 37 37 39 39 43 43 58 58 60 60 or 80 80 resp 106 106 or 114 114 then c g1 c g1 and c g2 c g2 are stably cc isomorphic with brnr c gi c2brnr c gi c2 explicit structures of non rational fields c g c g are given for each cases including also the case 30 30 with brnr c g c2 c2brnr c g c2 c2

graded polynomial identities of matrices we consider g graded polynomial identities of the p p matrix algebra mp k over a field k of characteristic 0 graded by an arbitrary group g we find relations between the g graded identities of the g graded algebra mp k and the g h graded identities of the tensor product of mp k and the h graded algebra mq k with a fine h grading we also find a basis of the g graded identities of mp k with an elementary grading such that the identity component coincides with the diagonal of mp k

the fully nonlinear development of gortler vortices in growing boundary layers the fully nonlinear development of small wavelength gortler vortices in a growing boundary layer is investigated by a combination of asymptotic and numerical methods the starting point for the analysis is the weakly nonlinear theory of hall j inst math applies 29 173 1982 who discussed the initial development of large wavenumber small amplitude vortices in a neighbourhood of the location where they first become linearly unstable that development is unusual in the context of nonlinear stability theory in that it is not described by the stuart watson approach in fact the development is governed by a pair of coupled nonlinear partial differential evolution equations for the vortex flow and the mean flow correction here the further development of this interaction is considered for vortices so large that the mean flow correction driven by them is as large as the basic state surprisingly it is found that such a nonlinear interaction can still be described by asymptotic means it is shown that the vortices spread out across the boundary layer and effectively drive the boundary layer in fact the system obtained by the equations for the fundamental component of the vortex generates a differential equation for the basic state thus the mean flow adjusts so as to make these large amplitude vortices locally neutral moreover in the region where the vortices exist the mean flow has a square root profile and the vortex velocity field can be written down in closed form the upper and lower boundaries of the region of vortex activity are determined by a free boundary problem involving the boundary layer equations in general it is found that this region ultimately includes almost all of the original boundary layer and much of the free stream in this situation the mean flow has essentially no relation to the flow that exists in the absence of the vortices