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

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

on fundamental solutions of generalized schrodinger operators abstract we consider the generalized schrodinger operator where is a nonnegative radon measure in r n n 3 assuming that satisfies certain scale invariant kato conditions and doubling conditions we establish the following bounds for the fundamental solution of in r n ce e 2 d x y x y n 2 x y ce e 1 d x y x y n 2 where d x y is the distance function for the modified agmon metric m x dx2 associated with we also study the boundedness of the corresponding riesz transforms 1 2 on lp r n dx

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

compactness of the solution operator to in weighted l2 spaces abstract in this paper we discuss compactness of the canonical solution operator to on weigthed l 2 spaces on c n for this purpose we apply ideas which were used for the witten laplacian in the real case and various methods of spectral theory of these operators we also point out connections to the theory of dirac and pauli operators

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

computing periodic orbits with arbitrary precision this paper deals with the computation of periodic orbits of dynamical systems up to any arbitrary precision these very high requirements are useful for example in the studies of complex pole location in many physical systems the algorithm is based on an optimized shooting method combined with a numerical ordinary differential equation ode solver tides that uses a taylor series method nowadays this methodology is the only one capable of reaching precision up to thousands of digits for odes the method is shown to be quadratically convergent some numerical tests for the paradigmatic lorenz model and the henon heiles hamiltonian are presented giving periodic orbits up to 1000 digits

on maximal sobolev and holder estimates for the tangential cauchy riemann operator and boundary laplacian let m be the boundary of a smoothly bounded pseudoconvex domain in c n n 3 or more generally any compact pseudoconvex cr manifold of dimension 2 n 1 for which the range of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is closed in l 2 in this article we study the l p sobolev and holder regularity properties of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i and b near a point of finite type under a comparable eigenvalues condition on the levi form we show that if all possible sums of q 0 eigenvalues of the levi matrix are comparable to its trace near a point of finite commutator type condition d q 0 then the inverse k q of b on 0 q forms for q 0 q n 1 q 0 satisfies sharp kernel estimates in terms of the quasi distance associated to the hormander sum of squares operator in particular we obtain the maximal l p estimates for b which were conjectured in the 1980s we also prove sharp estimates for certain parts of the kernels of k q 0 1 and k n q 0 and give some applications concerning domains with at most one degenerate eigenvalue finally we establish the composition and mapping properties of a class of singular integral nonisotropic smoothing operators that arises naturally in complex analysis these results yield optimal regularity of k q and related operators in the sobolev and lipschitz norms both isotropic and nonisotropic

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

triangular witt groups part ii from usual to derived we establish that the derived witt group is isomorphic to the usual witt group when 2 is invertible this key result opens the ali baba s cave of triangular witt groups linking the abstract results of part i to classical questions for the usual witt group for commercial purposes we survey the future applications of triangular witt groups in the introduction we also establish a connection between odd indexed witt groups and formations finally we prove that over a commutative local ring in which 2 is a unit the shifted derived witt groups are all zero but the usual one

on the behavior of the lorenz equation backward in time abstract the sets of solutions to the lorenz equations that exist backward in time and are bounded at an exponential rate determined by the eigenvalues of the linear part of the equation are examined the set associated with the middle eigenvalue is shown to project surjectively onto a plane thereby providing a lower estimate for its dimension specific bounds are also found for a cone containing this set

pairings in triangular witt theory abstract given a product k l m between triangulated categories with duality we show that under some conditions there exist naturally two different pairings w i k w j l w i j m where w denotes the triangulated witt functor of balmer p balmer k theory 19 2000 311 363 our main example of such a situation is the case that k l m is the bounded derived category of vector bundles over a scheme x and is the derived tensor product the derived witt groups of this scheme w x i z w i x become a graded skew commutative ring with two different product structures which are both equally natural in the last section we prove then a projection formula for our product and show as an application that a springer type theorem is true for the derived witt groups too

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

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

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

estimates for the bergman and szego kernels in mathbf c 2 the purpose of this paper some of whose conclusions were announced in nrsw is to study the bergman and szegd projection operators on pseudoconvex domains q of finite type in c2 the results we obtain are of three kinds i the size estimates of the bergman and szeg6 kernels and their derivatives ii the cancellation properties of those kernels expressed in terms of the actions of these operators on suitable bump functions these properties have an interest in their own right but are also crucial in iii below iii the sharp mapping properties of these operators on functions spaces such as lp the nonisotropic sobolev spaces and the nonisotropic lipschitz spaces which are naturally attached to the geometry of the boundary this also leads ultimately to sharp results in the regularity properties of solutions of the equation du f in q and 9bu f on the boundary let us describe these matters in more detail our starting points are certain basic geometric constructs associated to q a metric p defined in terms of the vector fields x1 and x2 which are the real and imaginary parts of the tangential levi vector field on d 2 and a function a p 8 a polynomial in 8 for p e d 2 which represents the higher levi invariant attached to d 9 the significance of these to the geometry of d q can be understood from the following facts the ball b p 8 c d q centered at p of radius 8 in the metric p when viewed in the appropriate coordinates has width 8 in the complex tangential directions but its width is a p 8 in the complementary direction thus the volume of the ball is 82a p 8 among other crucial properties

the fractal property of the lorenz attractor in a 1963 paper lorenz inferred that the lorenz attractor must be an infinite complex of surfaces we investigate this fractal property of the lorenz attractor in two ways firstly we obtain explicit plots of the fractal structure of the lorenz attractor using symbolic dynamics and multiple precision computations of periodic orbits the method we derive for multiple precision computation is based on iterative refinement and can compute even highly unstable periodic orbits with long symbol sequences with as many as 100 digits of accuracy ordinary numerical integrations are much too crude to show even the coarsest splitting of surfaces and there appear to be no other explicit computations of the fractal structure in the extensive literature about the lorenz attractor secondly we apply a well known formula that gives the hausdorff dimension of the lorenz attractor in terms of the characteristic multipliers of its unstable periodic orbits the formula converges impressively and the hausdorff dimension of the lorenz attractor appears to be 2 0627160 we use comparison with explicit computations of the fractal structure and discuss the accuracy of this formula and its applicability to the lorenz equations additionally we apply periodic orbit theory to the lorenz attractor and exhibit its spectral determinant and compute its lyapunov exponent

heat equations in r c abstract let p c r be a subharmonic nonharmonic polynomial and r a parameter define z p z p z a closed densely defined operator on l 2 c if p z p z p and 0 we solve the heat equation u s p u 0 u 0 z f z on 0 c the solution comes via the heat semigroup e s p and we show that u s z e s p f z c h p s z w f w d w we prove that h p is c off the diagonal s z w s 0 and z w and that h p and its derivatives have exponential decay in particular we give new estimates for the long time behavior of the heat equation

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

integrating the nonintegrable analytic structure of the lorenz system revisited abstract a study of the complex time analytic structure of the lorenz system in nonintegrable parameter regimes reveals the special sets of parameter values for which one time dependent integral of motion exists furthermore the analysis yields the exact form of the part of the integral with the highest homogeneous weight and a method to construct the rest of the integral recursive clustering of singularities in the chaotic regimes of the system is observed in computer studies and explained by a simple analytic argument the analytic techniques used in these studies a systematic resummation of a logarithmic psi series appears to be quite general and can provide explicit representations of a solution even in the chaotic regimes in the neighborhood of a given movable singularity furthermore we suggest that this technique provides a type of renormalization program to study a wide class of nonintegrable systems

complex singularities and the lorenz attractor the lorenz attractor is one of the best known examples of applied mathematics however much of what is known about it is a result of numerical calculations and not of mathemat ical analysis as a step toward mathematical analysis we allow the time variable in the three dimensional lorenz system to be complex hoping that solutions that have resisted analysis on the real line will give up their secrets in the complex plane knowledge of singularities being fundamental to any investigation in the complex plane we build upon earlier work and give a complete and consistent formal development of complex singular ities of the lorenz system using the psi series the psi series contain two undetermined constants in addition the location of the singularity is undetermined as a consequence of the autonomous nature of the lorenz system we prove that the psi series converge using a technique that is simpler and more powerful than that of hille thus implying a two parameter family of singular solutions of the lorenz system we pose three ques tions answers to which may bring us closer to understanding the connection of complex singularities to lorenz dynamics

pointwise estimates for relative fundamental solutions of heat equations in mathbb r times mathbb c let p mathbb c to mathbb r be a subharmonic nonharmonic polynomial and tau in mathbb r a parameter define bar z tau p frac partial partial bar z tau frac partial p partial bar z e tau p frac partial partial bar z e tau p a closed densely defined operator on l 2 mathbb c if square tau p bar z tau p bar z tau p and square tau p bar z tau p bar z tau p we solve the heat equations partial s u tilde square tau p u 0 u 0 z f z and partial s tilde u tilde square tau p tilde u 0 tilde u 0 z tilde f z we write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels we prove that the kernels are c off of the diagonal s z w s 0 and z w and find pointwise bounds for the kernels and their derivatives

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

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

pointwise estimates of weighted bergman kernels in several complex variables abstract we prove new pointwise bounds for weighted bergman kernels in c n whenever a coercivity condition is satisfied by the associated weighted kohn laplacian on 0 1 forms our results extend the ones obtained by christ in 5 our main idea is to develop a version of agmon theory originally introduced in 1 to deal with schrodinger operators for weighted kohn laplacians on 0 1 forms inspired by the fact that these are unitarily equivalent to certain generalized schrodinger operators

the algebraic approximation of attractors the finite dimensional case abstract in this article we present a constructive method for approximating attractors the approximate sets are parts of algebraic sets i e the zeros of high degree polynomial functions and they can approximate the attractor at an arbitrary high level of accuracy

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

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

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

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

a borel transform method for locating singularities of taylor and fourier series given a taylor series with a finite radius of convergence its borel transform defines an entire function a theorem of polya relates the large distance behavior of the borel transform in different directions to singularities of the original function with the help of the new asymptotic interpolation method of van der hoeven we show that from the knowledge of a large number of taylor coefficients we can identify precisely the location of such singularities as well as their type when they are isolated there is no risk of getting artefacts with this method which also gives us access to some of the singularities beyond the convergence disk the method can also be applied to fourier series of analytic periodic functions and is here tested on various instances constructed from solutions to the burgers equation large precision on scaling exponents up to twenty accurate digits can be achieved

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