decomposition of matrices into three involutions abstract in this paper we generalize some results of c s ballantine concerning products of three n n complex involutions we prove that each n n complex matrix a with determinant 1 and dimker a n 2 for all c is the product of three involutions on the other hand we show that if an n n complex matrix a is the product of three involutions then m 2n r 3 and m 3n 4 where m dimker a and r dimker a 3 for any 0 and 4 1 we also completely characterize products of three 5 5 complex involutions

impacts of noise on a class of partial differential equations abstract this paper is concerned with effects of noise on the solutions of partial differential equations we first provide a sufficient condition to ensure the existence of a unique positive solution for a class of stochastic partial differential equations then we prove that noise could induce singularities finite time blow up of solutions finally we show that a stochastic allen cahn equation does not have finite time singularities and the unique solution exists globally

structure of chevalley groups over rings via universal localization abstract in the current article we study the structure of a chevalley group g r over a commutative ring r we generalize and improve the following results on the standard relative and multi relative commutator formulas the nilpotent structure of relative k 1 the bounded word length of commutators to this end we enlarge the elementary group construct a generic element for the extended elementary group and use localization in the universal ring the key step is a construction of a generic element for the principal congruence subgroup corresponding to a principal ideal

products of two unipotent matrices of index 2 abstract necessary and sufficient conditions are presented for a square matrix a over a general field f to be the product of two unipotent matrices of index 2 this generalizes a result established by wang and wu 1991 4 for the case where f is the complex field

injective stability for k1 of the orthogonal group abstract the injective stabilization bound of l n vaserstein for k1 of the orthogonal group over an affine algebra over a perfect c 1 field is discussed

strong euler class of a stably free module of odd rank abstract let r be a commutative noetherian ring of dimension n 3 following a suggestion of fasel we establish a group homomorphism from van der kallen s group um n 1 r e n 1 r to the n th euler class group e n r so that 1 when n is even coincides with the homomorphism given by bhatwadekar and sridharan through euler classes 2 when n is odd is non trivial in general for an important class of rings 3 the sequence um n 1 r e n 1 r e n r e 0 n r 0 is exact where e 0 n r is the n th weak euler class group if x spec r is a smooth affine variety of dimension n over r so that the complex points of x are complete intersections and the canonical module k r is trivial then the sequence is proved to be exact on the left as well more generally let r be a commutative noetherian ring of dimension d and n be an integer such that n d 2 n 3 we also indicate how to extend our arguments to this setup to obtain a group homomorphism from um n 1 r e n 1 r to e n r

sums and products of two quadratic matrices abstract necessary and sufficient conditions are given for a complex matrix t to be the sum of an idempotent and a square zero matrix these conditions are expressed in terms of 1 the block structure of t under similarity as well as 2 the elementary divisor structure of t this leads to a complete characterization of which complex matrices are a sum of two quadratic matrices also using a similar scheme we characterize when a matrix is the product of an involution and a unipotent matrix of index 2

operations on orbits of unimodular vectors abstract taking the m power of an entry is a well defined operation on the unimodular vectors in an modulo addition operations if n is at least 3 for an arbitrary commutative ring a and any integer m

sobolev differentiable stochastic flows for sdes with singular coefficients applications to the transport equation in this paper we establish the existence of a stochastic flow of sobolev diffeomorphisms r d x 7 s t x r d s t r for a stochastic differential equation sde of the form dxt b t xt dt dbt s t r xs x r d the above sde is driven by a bounded measurable drift coefficientb r r d r d and a d dimensional brownian motion b more specifically we show that the stochastic flow s t of the sde lives in the space l 2 w 1 p r d w for all s t and all p 1 where w 1 p r d w denotes a weighted sobolev space with weight w possessing a p th moment with respect to lebesgue measure on r d this result is counter intuitive since the dominant culture in stochastic and deterministic dynamical systems is that the flow inherits its spatial regularity from the driving vector fields the spatial regularity of the stochastic flow yields existence and uniqueness of a sobolev differentiable weak solution of the stratonovich stochastic transport equation dtu t x b t x du t x dt p d 1 ei du t x db i t 0 u 0 x u0 x where b is bounded and measurable u0 is c 1 b and ei d 1 a basis for r d it is well known that the deterministic counter part of the above equation does not in general have a solution using stochastic perturbations and our analysis of the above sde we establish a deterministic flow of sobolev diffeomorphisms for classical one dimensional deterministic ode s driven by discontinuous vector fields furthermore and as a corollary of the latter result we construct a sobolev stochastic flow of diffeomorphisms for one dimensional sde s driven by discontinuous diffusion coefficients

generalized umbrella rings in this paper the notion of generalized umbrella rings is introduced we show that every generalized umbrella ring is a stably coherent ring as an application we show that every finitely generated projective module over r x is free if r is a generalized umbrella ring with weak global dimension two

on the effect of a noise on the solutions of the focusing supercritical nonlinear schrodinger equation we investigate the influence of a random perturbation of white noise type on the finite time blow up of solutions of a focusing supercritical nonlinear schrodinger equation we prove that contrary to the deterministic case any initial data gives birth to a solution which develops singularities moreover the singularities appear immediately we use a stochastic generalization of the variance identity and a control argument

strong solutions of stochastic equations with singular time dependent drift we prove existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time we only assume local l q l p integrability of b in g with d p 2 q 1 we also prove strong feller properties in this case if b is the gradient in x of a nonnegative function blowing up as g x g we prove that the conditions 2d t k 2d t ke 0 2 imply that the explosion time is infinite and the distributions of the solution have sub gaussian tails

homotopy and commutativity principle abstract we give evidence for the principle that a special linear symplectic orthogonal matrix over a commutative ring which is homotopic to the identity will commute up to an elementary matrix with all elements of the respective group

powers of 1 cyclic conjugacy classes in the groups gln f and sln f abstract let a gl n f where f is a field we say that a is 1 cyclic if a is similar to a matrix of the form a diag a 1 a 2 a k where a i gl l i f is cyclic for 1 i k l 1 0 1 and l i 2 for 2 i k it is shown that if a gl n f is 1 cyclic where n 2 and f 4 then every nonscalar matrix m gl n f whose determinant equals det a 4 is the product of four matrices which are similar to a under matrices of sl n f the problem of expressing a scalar matrix as a product of similar 1 cyclic matrices is also discussed the above result is applied to problems of factorizing matrices in the group sl n f into products of unipotent matrices of index 2 and into products of matrices of fixed finite order

a note on projective modules over real affine algebras let a be an affine domain over r of dimension d let f a be an element not belonging to any real maximal ideal of a and let p be a projective a module of rank d 1 let a p af pf be a unimodular element and q af pf a p af if p is free then a result of ojanguren and parimala 6 theorem shows that q is extended from a a consequence of this result is that if d 3 then all projective modules over af are free where a r x1 x2 x3 see 6 for motivation in this paper we prove the following result 3 10 which is a generalization of the above result of ojanguren and parimala

algebraic geometry of topological spaces i we use techniques from both real and complex algebraic geometry to study k theoretic and related invariants of the algebra c x of continuous complex valued functions on a compact hausdorff topological space x for example we prove a parameterized version of a theorem by joseph gubeladze we show that if m is a countable abelian cancellative torsion free semi normal monoid and x is contractible then every finitely generated projective module over c x m is free the particular case m mathbb n 0 n gives a parameterized version of the celebrated theorem proved independently by daniel quillen and andrei suslin that finitely generated projective modules over a polynomial ring over a field are free the conjecture of jonathan rosenberg which predicts the homotopy invariance of the negative algebraic k theory of c x follows from the particular case m mathbb z n we also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on c algebras and for a homology theory of commutative algebras to vanish on c algebras these criteria have numerous applications for example the vanishing criterion applied to nil k theory implies that commutative c algebras are k regular as another application we show that the familiar formulas of hochschild kostant rosenberg and loday quillen for the algebraic hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic hochschild and cyclic homology of c x applications to the conjectures of be linson soule and farrell jones are also given

factorization of matrices into quadratic ones iii abstract it is shown that every complex n n matrix t is the product of four quadratic matrices moreover if t is invertible then the number of required quadratic matrices can be reduced to three and three is sharp for n 4 complex matrices which are expressible as the product of an idempotent and an invertible quadratic matrix are also characterized

a nice group structure on the orbit space of unimodular rows ii abstract we establish an excision type theorem for niceness of group structure on the orbit space of unimodular rows of length n modulo elementary action this permits us to establish niceness for relative versions of results for the cases when n d 1 d being the dimension of the base algebra we then study and establish niceness for the case when n d and also establish a relative version when the base ring is a smooth affine algebra over an algebraically closed field

stably free modules in su prob 3 suslin had asked the following question let a be any affine algebra of dimension n over an algebraically closed field what is the smallest integer m such that all stably free projective modules of rank bigger than m are free all the examples in the literature of stably free non free modules have rank less than or equal to n 1 2 the aim of this note is to construct examples of such modules of large rank we construct rank p stably free non free modules over p 2 dimensional affine algebras over algebraically closed fields wherep is any prime these varieties are actually smooth and rational over c these are trivial as holomorphic vector bundles forp 2 this is classical forp 2 see ms so these are strictly algebraic examples i had described this construction in mk 1 and proved the result for p 2 we will reproduce the construction with necessary modifications in this note let p be any prime number and k any field letf x be any polynomial of degree p over k letf 0 a e k and fi xo xl f xo x1 xp f x0 x1 also let

a cohomological classification of vector bundles on smooth affine threefolds we give a cohomological classification of vector bundles of rank 2 on a smooth affine threefold over an algebraically closed field having characteristic unequal to 2 as a consequence we deduce that cancellation holds for rank 2 vector bundles on such varieties the proofs of these results involve three main ingredients first we give a description of the first nonstable a1 homotopy sheaf of the symplectic group second these computations can be used in concert with f morel s a1 homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques stemming from a version of the postnikov tower in a1 homotopy theory to reduce the classification results to cohomology vanishing statements third we prove the required vanishing statements

well posedness of semilinear stochastic wave equations with holder continuous coefficients abstract we prove that semilinear stochastic abstract wave equations including wave and plate equations are well posed in the strong sense with an holder continuous drift coefficient if 2 3 1 the uniqueness may fail for the corresponding deterministic pde and well posedness is restored by adding an external random forcing of white noise type this shows a kind of regularization by noise for the semilinear wave equation to prove the result we introduce an approach based on backward stochastic differential equations we also establish regularizing properties of the transition semigroup associated to the stochastic wave equation by using control theoretic results

on completing unimodular polynomial vectors of length three it is shown that if r is a local ring of dimension three with r then a polynomial three vector vq x vx x v2 x over r x can be completed to an invertible matrix if and only if it is unimodular in particular if 1 3 j then every stably free projective r xx y module is free

sums of commuting square zero transformations abstract we show that a linear transformation on a vector space is a sum of two commuting square zero transformations if and only if it is a nilpotent transformation with index of nilpotency at most 3 and the codimension of im t ker t in ker t is greater than or equal to the dimension of the space im t 2 we also characterize products of two commuting unipotent transformations with index 2

on triviality of the euler class group of a deleted neighbourhood of a smooth local scheme let k be an infinite perfect field and let x specr where r is the localization of a k algebra b of finite type with respect to a regular prime ideal p specb letd be a divisor onx with a decompositiond i 1di into its irreducible components di such that each di is nonsingular and they intersect transversally consider y x d a deleted neighbourhood of the closed point x0 of x let e be a vector bundle on y in this set up e is trivial in the following cases i dim x 3 gabber ga1 ii t 1 bhatwadekar rao br iii ranke min dimr 2 t rao ra1 the above exposition is borrowed from a paper of nisnevich ni which contains some beautiful applications of these purity theorems in the same set up now let y y be a closed point and let my be the corresponding maximal ideal of oy y it is natural to ask whether my is a complete intersection observing that the conormal bundle my my is trivial we consider a more general situation let i be the defining ideal of a closed subscheme z of y such that the conormal bundle i i2 defined on z is trivial is i a complete intersection the questions we are addressing in this paper are a little stronger in nature can a given basis of i i2 be lifted to a minimal set of generators of i we remark that with the hypothesis on x each component di is defined by a single equation fi which vanishes at the closed point x0 of x let m be the maximal ideal at x0 the condition of transversality of intersection of di s is equivalent to saying that the images of f1 ft in the a m vector space m m2 are linearly independent keeping this discussion in mind we now give an excerpt of our main results in commutative algebraic terms see 4 2 4 3 4 4

well posedness of the transport equation by stochastic perturbation we consider the linear transport equation with a globally holder continuous and bounded vector field with an integrability condition on the divergence while uniqueness may fail for the deterministic pde we prove that a multiplicative stochastic perturbation of brownian type is enough to render the equation well posed this seems to be the first explicit example of a pde of fluid dynamics that becomes well posed under the influence of a multiplicative noise the key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of ito tanaka type

construction of strong solutions of sde s via malliavin calculus abstract in this paper we develop a new method for the construction of strong solutions of stochastic equations with discontinuous coefficients we illustrate this approach by studying stochastic differential equations driven by the wiener process using malliavin calculus we derive the result of a k zvonkin 1974 31 for bounded and measurable drift coefficients as a special case of our analysis of sde s moreover our approach yields the important insight that the solutions obtained by zvonkin are even malliavin differentiable the latter indicates that the nature of strong solutions of sde s is tightly linked to the property of malliavin differentiability we also stress that our method does not involve a pathwise uniqueness argument but provides a direct construction of strong solutions

multiple commutators of elementary subgroups end of the line abstract in our previous joint papers with roozbeh hazrat and alexei stepanov we established commutator formulas for relative elementary subgroups in gl n r n 3 and other similar groups such as bak s unitary groups or chevalley groups in particular there it was shown that multiple commutators of elementary subgroups can be reduced to double such commutators however since the proofs of these results depended on the standard commutator formulas it was assumed that the ground ring r is quasi finite here we propose a different approach which allows to lift any such assumptions and establish almost definitive results in particular we prove multiple commutator formulas and other related facts for gl n r over an arbitrary associative ring r

well posedness of sdes with drifts in mixed norm spaces and driven by mixed noises abstract we prove the strong well posedness for stochastic differential equations driven by partial stable noise and partial brownian noise the drift coefficients belong to certain mixed norm spaces which generalize the known results even for stochastic differential equations with single noise cases

the effect of a noise on the stochastic modified camassa holm equation we study the effect of noise on the stochastic modified camassa holm equation we first derive the stochastic modified camassa holm equation by the stochastic variational principle then we prove the well posedness of the stochastic modified camassa holm equation by the iterative process under the condition of the small noise intensity we can also get the regularization of the solution from the parabolic term

odd orthogonal matrices and the non injectivity of the vaserstein symbol abstract r a rao w van der kallen showed that the vaserstein symbol v s r 3 from the orbit space of unimodular rows of length three over the coordinate ring of the real three sphere s r 3 modulo elementary action to the elementary symplectic witt group w e s r 3 is not injective dhvanita r rao neena gupta gave an uncountable family of singular real threefolds a for which the vaserstein symbol v a is not injective in this paper we give a countable family of smooth real birationally equivalent threefolds a n for which the vaserstein symbol v a n is not injective

on invariance of the euler class groups under a subintegral base change abstract let r be a commutative noetherian ring and r s be a subintegral extension it is proved that the top euler class group of r is isomorphic to the top euler class group of s an example is given to show that similar conclusion does not hold for integral but not subintegral extension of rings

difference and similarity models of two idempotent operators in a recent work hartwig and putcha obtained a complete characterization of those finite matrices which can be expressed as the difference of two idempotents extending this result to operators on a possibly infinite dimensional hilbert space seems more difficult in this paper we initiate its study and obtain among other things 1 that not every nilpotent operator is the difference of two idempotents 2 that if t is the difference of two idempotents then the spectra of t and t differ at most by the two points k 1 and 3 a characterization of differences of two idempotents among normal operators in the second part of the paper we develop some similarity invariant models of two idempotents these are analogous to the known unitary equivalence invariant models for two orthogonal projections

excision in algebraic k theory introduction 1 excision and the homology of affine groups 2 volodin s spaces 3 excision for rings with the triple factorization property 4 lie analogue of volodin s construction 5 the homology of nilpotent groups and nilpotent lie algebras 6 a proof of the main theorems 7 the tensor product of h unital algebras 8 the matrix ring of a small category 9 the acyclicity of triangular complexes 10 a proof of karoubi s conjecture

product of invertible operators of quadratic type abstract suppose all invertible quadratic operators t are assumed to satisfy t 2 bt i 0 we show that a complex matrix t is the product of two invertible quadratic matrices if and only if t is similar to a matrix of the form d d 1 i n i m i 1 j i 2 i 2 i 2 i xy 2 i yx 1 where b b 2 4 2 0 1 and gb 2 are not eigenvalues of d n xy and yx are nilpotent and each j i is a nilpotent jordan block of even size also we show that an n n matrix t is the product of finitely many invertible quadratic matrices if and only if det t m where m is an integer for n odd and is an even integer for n even on the other hand for operators on an infinite dimensional hilbert space we characterize the products of two and four invertible quadratic operators among normal operators and show that every invertible operator is the product of six invertible quadratic operators

sums of two triangularizable quadratic matrices over an arbitrary field abstract let k be an arbitrary field and a b c d be elements of k such that the polynomials t 2 at b and t 2 ct d are split in k t given a square matrix m m n k we give necessary and sufficient conditions for the existence of two matrices a and b such that m a b a 2 aa bi n and b 2 cb di n prior to this paper such conditions were known in the case b d 0 a 0 and c 0 4 and in the case a b c d 0 1 here we complete the study which essentially amounts to determining when a matrix is the sum of an idempotent and a square zero matrix this generalizes results of wang 5 to an arbitrary field possibly of characteristic 2

an abelian group structure on orbits of unimodular squares in dimension 3 the subject of whether orbit sets of unimodular rows modulo the elementary subgroup um a re a have a group structure probably r r w x began first in 9 with the vaserstein symbol v um a re a a w a 3 3 e which was shown to be bijective when the stable dimension of a was less than or equal to 2 here the stable dimension of the ring a is in the sense of bass and equals the stable rank of a y 1 the reader could replace it by the more familiar krull dimension for simplicity the vaserstein symbol v and the grothendieck group w a of stably equivalent alternating e w x matrices of pfaffian 1 is described in 9 p 945 and briefly in the next section w x in 1 2 van der kallen has shown that if a is a commutative ring of stable dimension d then the coset space um a re a has the structure n n 1 4 of an abelian group provided n g max 3 d q 2 for topological situa2 tions like a being the ring of continuous functions c x r on a finite cw complex x of dimension d he has given some nice models namely the ny1 higher cohomotopy groups p x for this group w x in 5 it was shown that the vaserstein symbol was not injective for three dimensional rings so one could not expect to get a very nice group structure on the orbit set of unimodular 3 vectors over a three dimensional ring in this article we will prove that the orbit set of squares of

on the non injectivity of the vaserstein symbol in dimension three abstract l n vaserstein proved over a two dimensional ring that the orbit space of unimodular rows of length three modulo elementary action has a witt group structure r a rao and w van der kallen showed that the vaserstein symbol need not be injective over three dimensional affine algebras over the real field but is injective over three dimensional affine algebras over a field of cohomological dimension one whose characteristic 2 3 r g swan r a rao and j fasel gave another example of a real affine algebra of dimension three for which the vaserstein symbol is not injective we demonstrate an uncountably infinite family of non isomorphic affine algebras of dimension three over the real field for which the vaserstein symbol is not injective

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

noise prevents singularities in linear transport equations a stochastic linear transport equation with multiplicative noise is considered and the question of no blow up is investigated the drift is assumed only integrable to a certain power opposite to the deterministic case where smooth initial conditions may develop discontinuities we prove that a certain sobolev degree of regularity is maintained which implies holder continuity of solutions the proof is based on a careful analysis of the associated stochastic flow of characteristics

numerical simulation of focusing stochastic nonlinear schrodinger equations abstract in this paper we numerically investigate nonlinear schrodinger equations with a stochastic contribution which is of white noise type and acts either as a potential multiplicative noise or as a forcing term additive noise in the subcritical case we recover similar results as in the case of the korteweg de vries equation in the critical or supercritical case we observe that depending on its smoothness the noise may have different effects spatially smooth noises amplify blow up phenomena whereas delta correlated multiplicative noises prevent blow up formation note that in this latter case very few results are known both from a theoretical and a numerical point of view

stably free modules over of rank dim are free we prove that for any finite dimensional ring r and n dim r 2 the group e n r x acts transitively on um n r x in particular we obtain that for any finite dimensional ring r all finitely generated stably free modules over r x of rank dim r are free this result was only known for noetherian rings the proof we give is short simple and constructive

the euler class group of a polynomial algebra with coefficients in a line bundle let a be a commutative noetherian ring and p be a projective a module of rank text dim a 1 an intriguing open question is to find the precise obstruction for p to split as p simeq q oplus a for some a module q in this paper we settle this question when a r t for some ring r containing the field of rationals and p is a projective a module of rank text dim r

rank equalities for idempotent matrices with applications we show by elementary block matrix operations a variety of rank equalities related to sums of two idempotent matrices and give various consequences and applications

products of involutions in the stable general linear group abstract in the stable general linear group over an arbitrary field we prove that every element with determinant 1 is the product of three involutions and of no less in general we also obtain several results of the same flavor with applications to decompositions of automorphisms of an infinite dimensional vector space that are scalar multiples of finite rank perturbations of the identity

a sufficient condition for the kolmogorov 4 5 law for stationary martingale solutions to the 3d navier stokes equations we prove that statistically stationary martingale solutions of the 3d navier stokes equations on mathbb t 3 subjected to white in time colored in space forcing satisfy the kolmogorov 4 5 law in an averaged sense and over a suitable inertial range using only the assumption that the kinetic energy is o nu 1 as nu rightarrow 0 where is the inverse reynolds number this plays the role of a weak anomalous dissipation no energy balance or additional regularity is assumed aside from that satisfied by all martingale solutions from the energy inequality if the force is statistically homogeneous then any homogeneous martingale solution satisfies the spherically averaged 4 5 law pointwise in space an additional hypothesis of approximate isotropy in the inertial range gives the traditional version of the kolmogorov law we demonstrate a necessary condition by proving that energy balance and an additional quantitative regularity estimate as nu rightarrow 0 imply that the 4 5 law or any similar scaling law cannot hold

the nonlinear schrodinger equation with white noise dispersion abstract under certain scaling the nonlinear schrodinger equation with random dispersion converges to the nonlinear schrodinger equation with white noise dispersion the aim of this work is to prove that this latter equation is globally well posed in l 2 or h 1 the main ingredient is the generalization of the classical strichartz estimates additionally we justify rigorously the formal limit described above

excision in algebraic k theory and karoubi s conjecture abstract we prove that the property of excision in algebraic k theory is for a q algebra a equivalent to the h unitality of the latter our excision theorem in particular implies karoubi s conjecture on the equality of algebraic and topological k theory groups of stable c algebras it also allows us to identify the algebraic k theory of the symbol map in the theory of pseudodifferential operators

blow up for the stochastic nonlinear schrodinger equation with multiplicative noise we study the influence of a multiplicative gaussian noise white in time and correlated in space on the blow up phenomenon in the supercritical nonlinear schrodinger equation we prove that any sufficiently regular and localized deterministic initial data gives rise to a solution which blows up in arbitrarily small time with a positive probability

regularization by noise for stochastic hamilton jacobi equations we study regularizing effects of nonlinear stochastic perturbations for fully nonlinear pde more precisely path by path l infty bounds for the second derivative of solutions to such pde are shown these bounds are expressed as solutions to reflected sde and are shown to be optimal

on maximal ideals in polynomial and laurent polynomial rings abstract when r is a complete local ring containing a field of characteristic zero we show that any maximal ideal in r t t 1 is projectively generated if r is a regular ring containing a field of characteristic zero we show that any maximal ideal m of r t such that ht m dim r is efficiently generated

sums of idempotent matrices abstract we show that any complex square matrix t is a sum of finitely many idempotent matrices if and only if trt is an integer and trt rank t moreover in this case the idempotents may be chosen such that each has rank one and has range contained in that of t we also consider the problem of the minimum number of idempotents needed to sum to t and obtain some partial results

products of unipotent matrices of index 2 abstract we show that an n n complex matrix t is the product of two unipotent matrices of index 2 if and only if t is similar to a matrix of the form d d 1 i n i mi 1 ji where 0 and 1 are not eigenvalues of d n is nilpotent and each ji is a nilpotent jordan block of even size on the other hand t is the product of finitely many unipotent matrices of index 2 if and only if det t 1 in this case the minimal number of required unipotents is 1 if n 1 3 if n 2 and 4 if n 3

the density of the solution to the stochastic transport equation with fractional noise we consider the transport equation driven by the fractional brownian motion we study the existence and the uniqueness of the weak solution and by using the tools of the malliavin calculus we prove the existence of the density of the solution and we give gaussian estimates from above and from below for this density

the hermite ring conjecture in dimension one abstract we prove constructively that for any ring r of krull dimension 1 and n 3 the group e n r x acts transitively on um n r x in particular we obtain that for any ring r with krull dimension 1 all finitely generated stably free modules over r x are free this settles the long standing hermite ring conjecture for rings of krull dimension 1

sobolev differentiable flows of sdes with local sobolev and super linear growth coefficients by establishing a characterization for sobolev differentiability of random fields we prove the weak differentiability of solutions to stochastic differential equations with local sobolev and super linear growth coefficients with respect to the starting point moreover we also study the strong feller property and the irreducibility to the associated diffusion semigroup

mathbb p 1 p 1 gluing for local complete intersections we prove an analogue of the affine horrocks theorem for local complete intersection ideals of height n in r t where r is a regular domain of dimension d which is essentially of finite type over an infinite perfect field of characteristic unequal to 2 and 2n ge d 3

chow witt groups and grothendieck witt groups of regular schemes let a be a noetherian commutative z 1 2 algebra of krull dimension d and let p be a projective a module of rank d we use derived grothendieck witt groups and euler classes to detect some obstructions for p to split off a free factor of rank one if d 3 we show that the vanishing of its euler class in the corresponding grothendieck witt group is a necessary and sufficient condition for p to have a free factor of rank one if d is odd we also get some results in that direction if a is regular we show that the chow witt groups defined by morel and barge appear naturally as some homology groups of a gersten type complex in grothendieck witt theory from this we deduce that if d 3 then the vanishing of the euler class of p in the corresponding chow witt group is a necessary and sufficient condition for p to have a free factor of rank one crown copyright c 2008 published by elsevier inc all rights reserved

on the spectra of some combinations of two generalized quadratic matrices let a and b be two generalized quadratic matrices with respect to idempotent matrices p and q respectively such that a p a p 0 a p p a a b q b q 0 b q q b b p q q p ab ba and a b p q p q a b with c let a b be diagonalizable the relations between the spectrum of the matrix a b and the spectra of some matrices produced from a and b are considered moreover some results on the spectrum of the matrix a b are obtained when a b is not diagonalizable finally some results and examples illustrating the applications of the results in the work are given

on the existence of unimodular elements and cancellation of projective modules over noetherian and non noetherian rings let r be a commutative ring of dimension d s r x or r x 1 x and p a finitely generated projective s module of rank r then p is cancellative if p has a unimodular element and r d 1 moreover if r dim s then p has a unimodular element and therefore p is cancellative as an application we prove that if r is a ring of dimension d of finite type over a prufer domain and p is a projective r x or r x 1 x module of rank at least d 1 then p has a unimodular element and is cancellative

the euler class group of a polynomial algebra in this paper we continue our study on the theory of the euler class group of a polynomial algebra a t where a is a commutative noetherian ring containing q of dimension n for such a ring a in 9 we defined the notion of the nth euler class group en a t of a t for simplicity let us call it e a t in 9 we also studied the relations between e a t and e a where e a is the nth euler class group of a for example there is canonical map e a e a t which is an injective group homomorphism and it is an isomorphism when a is a smooth affine domain 9 proposition 5 7 in general these two groups are not isomorphic see discussion preceding 9 proposition 5 7 in this context the following question is natural