regularizations of residue currents under assumptions about complete intersection we prove that coleff herrera type currents satisfy a robust calculus in the sense that natural regularizations of such currents can be multiplied to yield regularizations of the coleff herrera product of the currents

regularizations of products of residue and principal value currents let f 1 and f 2 be two functions on some complex n manifold and let be a test form of bidegree n n 2 assume that f 1 f 2 defines a complete intersection the integral of f 1 f 2 on f 1 2 e lunate 1 f 2 2 e lunate 2 is the residue integral i f1 f 2 e lunate 1 e lunate 2 it is in general discontinuous at the origin let 1 and 2 be smooth functions on 0 such that j 0 0 and j 1 we prove that the regularized residue integral defined as the integral of over 1 over 2 f 1 f 2 where j j f j 2 e lunate j is holder continuous on the closed first quarter and that the value at zero is the coleff herrera residue current acting on in fact we prove that if is a test form of bidegree n n 1 then the integral of 1 over 2 f 1 f 2 is holder continuous and tends to the over potential 1 f 1 over 1 f 2 of the coleff herrera current acting on more generally let f 1 and f 2 be sections of some vector bundles and assume that f 1 f 2 defines a complete intersection there are associated principal value currents u f and u g and residue currents r f and r g the residue currents equal the coleff herrera residue currents locally one can give meaning to formal expressions such as e g u f r g in such a way that formal leibnitz rules hold our results generalize to products of these currents as well 2006 elsevier inc all rights reserved

hua operators poisson transform and relative discrete series on line bundles over bounded symmetric domains let d g k be a bounded symmetric domain and s k l be its shilov boundary we consider the action of g with weight nu in mathbb z on functions on d viewed as sections the line bundle and the corresponding eigenspace of g invariant di erential operators the poisson transform maps hyperfunctions on the s to the eigenspaces we characterize the image in terms of hua operators on the sections of the line bundle for some special parameter the poisson transform is of szego type mapping into the relative discrete series we compute the corresponding elements in the discrete series

shimura invariant differential operators and their eigenvalues let bar d be the covariant cauchy riemann operator and mathcal d the covariant holomorphic differential operator on a line bundle over a hermitian symmetric space g k we study the shimura invariant differential operators defined via bar d and mathcal d we find the eigenvalues of a family of the shimura operators and of the generators

explicit serre duality on complex spaces abstract in this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization as well as a new analytic proof of serre duality on any reduced pure n dimensional paracompact complex space x at the core of the paper is the introduction of certain fine sheaves b x n q of currents on x of bidegree n q such that the dolbeault complex b x n becomes in a certain sense a dualizing complex in particular if x is cohen macaulay then b x n is an explicit fine resolution of the grothendieck dualizing sheaf

recent improvements in the complexity of the effective nullstellensatz we bring up to date the estimates on the complexity of the effective nullstellensatz and the membership problem

abelian differentials on singular varieties and variations on a theorem of lie griffiths we compare various notions of meromorphic and holomorphic differential forms on singular analytic varieties in particular we find that every meromorphic form gives rise to a canonical principal value current with support in the variety demanding that this current be closed we obtain a useful analytic description of the dualizing sheaf we then go on to generalize the lie griffiths theorem on one hand it is shown that a rational trace is enough to ensure algebraicity of the data on the other hand we prove that zero trace implies that the form is holomorphic in the sense of currents in fact we prove a more general local version of the theorem

hilbert series of nearly holomorphic sections on generalized flag manifolds abstract let x g p be a complex flag manifold and e x be a g homogeneous holomorphic vector bundle fix a u invariant kahler metric on x with u g maximal compact we study the sheaf of nearly holomorphic sections and show that the space of global nearly holomorphic sections in e coincides with the space of u finite smooth sections in e the degree of nearly holomorphic sections defines a u invariant filtration on this space using sheaf cohomology we determine in suitable cases the corresponding hilbert series it turns out that this is given in terms of lusztig s q analog of kostant s weight multiplicity formula and hence gives a new representation theoretic interpretation of these formulas

long range order with local chaos in lattices of diffusively coupled odes abstract the existence of non trivial collective behavior in lattices of diffusively coupled differential equations is investigated for a two dimensional square lattice of rossler systems a rotating long range order is observed this case is best described in terms of a complex ginzburg landau cgl equation submitted to the local noise produced by the chaotic rossler units the parameters of this cgl equation are estimated to be in the so called benjamin feir stable region the collective oscillation regime thus corresponds to the linearly stable spatially homogeneous solution of the equivalent cgl equation the possibility of more complex collective behavior in similar systems is discussed

ideals of smooth functions and residue currents abstract let f f 1 f m be a holomorphic mapping in a neighborhood of the origin in c n we find sufficient condition in terms of residue currents for a smooth function to belong to the ideal in c or ck generated by f if f is a complete intersection the condition is necessary more generally we give a sufficient condition for an element of class c or ck in the koszul complex induced by f to be exact for the proofs we introduce explicit homotopy formulas for the koszul complex induced by f

various approaches to products of residue currents abstract we describe various approaches to coleff herrera products of residue currents r j of cauchy fantappie leray type associated to holomorphic mappings f j more precisely we study to which extent exterior products of natural regularizations of the individual currents r j yield regularizations of the corresponding coleff herrera products our results hold globally on an arbitrary pure dimensional complex space

residue currents with prescribed annihilator ideals on singular varieties given an ideal mathcal j on a complex manifold andersson and wulcan constructed explicitly a current r mathcal j such that the annihilator of r mathcal j is mathcal j generalizing the duality theorem for coleff herrera products we describe a way to generalize this construction to ideals on singular varieties

integral formulas and an explicit version of the fundamental principle soit p c n c m une application polynomiale et on considere le systeme d equations differentielles p j d f t 0 j 1 m ou d i t et f est une fonction lisse dans un domaine convexe borne r n on presente une version explicite qui emploie des formules integrales et des courants residus du principe fondamental de ehrenpreis et palamodov

decomposition of residue currents given a submodule j o r 0 and a free resolution of j one can de ne a certain vector valued residue current whose annihilator is j we make a decomposition of the current with respect to assj that corresponds to a primary decomposition of j as a tool we introduce a class of currents that includes usual residue and principal value currents in particular these currents admit a certain type of restriction to analytic varieties and more generally to constructible sets

the plancherel formula for spherical functions with a one dimensional k type on a simply connected simple lie group of hermitian type abstract we generalize the harish chandra inversion formula for the spherical transform on a riemannian symmetric space to homogeneous line bundles on a hermitian symmetric space we determine the plancherel measure explicitly

on explicit solutions to the bezout equation explicit constructions are given for the solution of the bezout equation for matrices whose coefficients are multivariate polynomials or entire functions of paley wiener type this procedure also provides the 1 inverse when the bezout equation has no solutions

residue calculus and effective nullstellensatz multivariate residue calculus in the spirit of j lipman is developed from the computational point of view for example with several variants of the classical transformation law and used in order to make totally explicit the bezout identity and therefore the algebraic nullstellensatz in k x 1 x n where k is an infinite field of arbitrary characteristic such identities provide sharp size estimates for the denominator and the divisors in the bezout identity when k is the quotient field of a factorial regular ring a equipped with a size such as z or f p 1 q the estimates obtained by the authors in a previous work in the case a z are sharpened here while analytic techniques are replaced with an algebraic approach

weighted koppelman formulas and the equation on an analytic space let x be an analytic space of pure dimension we introduce a formalism to generate intrinsic weighted koppelman formulas on x that provide solutions to the partial derivative over bar equation we obtain new existence results for the partial derivative over bar equation as well as new proofs of various known results c 2011 elsevier inc all rights reserved

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

hua operators and poisson transform for bounded symmetric domains let be a bounded symmetric domain of non tube type in c n with rank r and s its shilov boundary we consider the poisson transform p s f z for a hyperfunction f on s defined by the poisson kernel p s z u h z z n r h z u n r 2 s z u s s c for all s satisfying certain non integral condition we find a necessary and sufficient condition for the functions in the image of the poisson transform in terms of hua operators when is the type i matrix domain in m n m c n less than or slanted equal to m we prove that an eigenvalue equation for the second order m n n valued hua operator characterizes the image 2006 elsevier inc all rights reserved

on the local global conjecture for integral apollonian gaskets we prove that a set of density one satisfies the local global conjecture for integral apollonian gaskets that is for a fixed integral primitive apollonian gasket almost every in the sense of density admissible passing local obstructions integer is the curvature of some circle in the gasket

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

koppelman formulas on flag manifolds and harmonic forms we construct koppelman formulas on manifolds of flags in mathbb c n for forms with values in any holomorphic line bundle as well as in the tautological vector bundles and their duals as an application we obtain new explicit proofs of some vanishing theorems of the bott borel weil type by solving the corresponding bar partial equation we also construct reproducing kernels for harmonic p q forms in the case of grassmannians

residue currents constructed from resolutions of monomial ideals given a free resolution of an ideal j of holomorphic functions one can construct a vector valued residue current r whose annihilator is precisely j in this paper we compute r in case j is a monomial ideal and the resolution is a cellular resolution in the sense of bayer and sturmfels a description of r is given in terms of the underlying polyhedral cell complex and it is related to irreducible decompositions of j

polynomial iteration in characteristic p abstract let f x s 0 d a s x s z x be a polynomial with a d 0 mod p take z f p and let o z f i z i z f p be the orbit of z under f where f i z f f i 1 z and f 0 z z for m o z we study the diameter of the partial orbit o z m z f z f 2 z f m 1 z and prove that diam o z m min m c log log m m p c m 1 2 p 1 2 where diameter is naturally defined in f p and c depends only on d for a complete orbit c we prove that diam c min p c e t 4 where t is the period of the orbit

residue current approach to ehrenpreis malgrange type theorem for linear differential equations with constant coefficients and commensurate time lags abstract we introduce a ring h of partial difference differential operators with constant coefficients initially defined by h glusing lursen for ordinary difference differential operators and investigate its cohomological properties combining this ring theoretic observation with the integral representation technique developed by m andersson we solve a certain type of division with bounds in the last section we deduce from this injectivity properties of various function modules over h as well as the density results of exponential polynomial solutions for partial difference differential equations

koppelman formulas on grassmannians we construct koppelman formulas on grassmannians for forms with values in any holomorphic line bundle as well as in the tautological vector bundle and its dual as an application we obtain new explicit proofs of some vanishing theorems of the bott borel weil type by solving the corresponding equation we also relate the projection part of our formulas to the bergman kernels associated to the line bundles

division formulas on projective varieties we introduce a division formula on a possibly singular projective subvariety x of complex projective space mathbb p n which e g provides explicit representations of solutions to various polynomial division problems on the affine part of x especially we consider a global effective version of the briancon skoda huneke theorem

boundary value problems for various boundaries of hermitian symmetric spaces abstract the image of the poisson transform on a principal series representations on a boundary component of a hermitian symmetric space is considered we prove that the image is characterized by a covariant differential operator on a homogeneous line bundle on the symmetric space

invariant differential operators on hermitian symmetric spaces our object of study is the ring of invariant differential operators on a hermitian symmetric space g k of classical and noncompact type here as usual g is a connected noncompact semisimple lie group with finite center and k is a maximal compact subgroup of g it is well known that the ring denoted by g k is isomorphic to a polynomial ring of 1 variables 1 being the rank of g k this is true even in the nonhermitian case if 1 1 it is generated by the laplace beltrami operator z which is essentially self adjoint moreover iz2 is nonnegative in the general case an easily posable problem is to find an explicitly defined set of generators we can go one step further by focusing our attention on the nonnegativity which necessarily limits the range of eigenvalues under a suitable integrability condition on functions it is thus natural to ask whether there exist some canonically defined operators which generate 9 g k and have the property of nonnegativity the main purpose of the present paper is to give an affirmative answer to this question in fact our answer applies to a somewhat more general type of rings of differential operators which includes 9 g k as a special case to be explicit we take an irreducible representation p k gl v with a complex vector space v of finite dimension and consider the set c p of all v valued cx functions f on g such that f xk 1 p k f x for every k e k we then denote by p the ring of left invariant difrrential operators on g which map c p into itself now the complexification g of the lie algebra of g has abelian subalgebras p and pwhich can be identified with the spaces of holomorphic and antiholomorphic tangent vectors on g k at the origin for any complex vector space w let sr w denote the ring of all complex valued homogeneous polynomial functions on w of degree r and let s w er 0sr w through the adjoint representation of g on g k acts naturally on sr it is a known fact due to hua and schmid that each irreducible constituent of this representation of k has multiplicity one now our principal

courants du type residuel attaches a une intersection complete abstract we construct in complete intersection s case elementary currents which describe the local ideal and give a decomposition in it for holomorphic function

soliton dynamics in a microstructured lattice model the nonlinear dynamics of localized structures of the soliton type in a lattice model involving internal degrees of freedom in rotation is presented the physical model consists basically of a one dimensional monoatomic chain equipped with microscopic electric dipoles associated with molecular groups the model is particularly suitable for molecular ferroelectric crystals such as sodium nitrite exhibiting structures in ferroelectric domains and walls or for long chains of polymers such as crystalline polyethylene or polyvinylidine fluoride and others where molecular groups perform rotational motions two main types of motion can be distinguished in this picture i the longitudinal and transverse displacements of the mass centre of the molecular group and ii two rigid body rotational motions of the electric dipoles about the chain axis and perpendicular to it the propagation of coupled solitons is investigated for two elementary configurations a situation for which all the dipoles rotate perpendicularly to the chain axis is first studied a second configuration is next considered when the dipole rotation takes place about the chain axis

boundary value problems for the shilov boundary of a bounded symmetric domain of tube type abstract in this article the images of the poisson transform on the degenerate series representations attached to the shilov boundary of a bounded symmetric domain of tube type are considered we characterize the images by means of second order differential equations

a global lojasiewicz inequality for algebraic varieties let x be the locus of common zeros of polynomials f 1 f in n complex variables a global upper bound for the distance to x is given in the form of a lojasiewicz inequality the exponent in this inequality is bounded by d min n k where d max 3 deg f i the estimates are also valid over an algebraically closed field of any characteristic

integral representation with weights ii division and interpolation let f be a r m matrix of holomorphic functions that is generically surjective we provide explicit integral representation of holomorphic such that f provided that is holomorphic and annihilates a certain residue current with support on the set where f is not surjective we also consider formulas for interpolation as applications we obtain generalizations of various results previously known for the case r 1

estimates of solutions of the h p and bmoa corona problem we prove new sharper estimates of solutions to the h p corona problem in strictly pseudoconvex domains in particular we show that the constant is independent of the number of generators we also obtain sharper estimates for solutions to the bmoa corona problem the proofs also lead to new results about the taylor spectrum of analytic toeplitz operators on h p and bmoa

sharp estimates for the arithmetic nullstellensatz we present sharp estimates for the degree and the height of the polynomials in the nullstellensatz over the integer ring zz the result improves previous work of philippon berenstein yger and krick pardo we also present degree and height estimates of intrinsic type which depend mainly on the degree and the height of the input polynomial system as an application we derive an effective arithmetic nullstellensatz for sparse polynomial systems the proof of these results relies heavily on the notion of local height of an affine variety defined over a number field we introduce this notion and study its basic properties

toeplitz operators on bounded symmetric domains in this paper jordan algebraic methods are applied to study toeplitz operators on the hardy space h2 s associated with the shilov boundary s of a bounded symmetric domain d in c of arbitrary rank the jordan triple system z c associated with d is used to determine the relationship between toeplitz operators and differential operators further it is shown that each jordan triple idempotent e e z induces an irreducible representation e symbol of the c algebra generated by all toeplitz operators tf with continuous symbol function f 0 introduction toeplitz operators on the boundary t aa of the open unit disc a c c play an important role in function theory of one complex variable cf 9 chapter 7 in several dimensions toeplitz operators have been mainly studied for strictly pseudo convex domains d c c 4 13 14 24 28 in particular for the hilbert ball 7 using the relationship with pseudo differential operators another class of domains in c generalizing the unit disc is the class of bounded symmetric domains cartan domains and exceptional domains which have a more complicated boundary structure compared to the strictly pseudo convex case the aim of this paper is the study of toeplitz operators t1 with continuous symbol functionf e c s on the shilov boundary s of a bounded symmetric domain d of arbitrary rank r in the special cases of the hilbert ball r 1 and the lie ball r 2 the structure of the operators ti and of the toeplitz c algebra

effective bezout identities in q z1 zn abstract if p sub 1 p sub m are n variate polynomials with integral coefficients and no common zeros in c exp n brownawell has shown in 1986 that there exist q sub 1 q sub m polynomials with integral coefficients and nu is an element of z such that p sub 1 q sub 1 p sub m q sub m nu and max deg q sub j max deg p sub j exp n on the other hand if h logarithm of the largest coefficient of all the p sub j and h sub 1 is the corresponding quantity for the q sub j then there is no sharp estimate of h sub 1 in terms of h and max deg p sub j in this paper we show that when the variety of common zeros at infinity of the p sub j is discrete then essentially we have h sub 1 d exp cn h for an absolute constant c if there were an algorithm to compute the q sub j in d exp cn time one would obtain exactly the above estimate current algorithms require about d exp n squared operations

integral representation with weights i we describe a new approach to representation formulas for holomorphic functions including the cauchy fantappie leray formula and provide a general method to generate weighted formulas

explicit representation of membership in polynomial ideals we introduce a new division formula on projective space which provides explicit solutions to various polynomial division problems with sharp degree estimates we consider simple examples as the classical macaulay theorem as well as a quite recent result by hickel related to the effective nullstellensatz we also obtain a related result that generalizes max noether s classical af bg theorem

function spaces and reproducing kernels on bounded symmetric domains there are several natural hilbert spaces of holomorphic functions on a bounded symmetric domain such are the bergman type spaces on which the holomorphic discrete series operates and the hardy type spaces which are related to the analytic continuation of the holomorphic discrete series also closely related is the bergmann space of entire functions on the ambient which arises as the closure of the polynomials with respect to a natural inner product the space of holomorphic polynomials decomposes into irreducible subspaces under the action of the isotropy group k of the domain the main facts about this decomposition were proved by schmid 21 for another proof see 22 each irreducible subspace contains a unique normalized l invariant spherical polynomial where l is the isotropy group of the shilov boundary in k our first main result is the explicit computation of the norms of the spherical polynomials with respect to each of the hilbert spaces considered for the domains of classical type they were considered by hua 9 and for some of the hilbert spaces this was done before by upmeier 24 using different methods for certain others there are partial results in 22 we are able to do this in a fairly simple unified way by making strong use of gindikin s generalized gamma function s next we obtain a description of the reproducing kernels of the k irreducible subspaces in each of our hilbert spaces and an expansion in terms of these for

a dolbeault grothendieck lemma on complex spaces via koppelman formulas let x be a complex space of pure dimension we introduce fine sheaves a x q of 0 q currents which coincides with the sheaves of smooth forms on the regular part of x so that the associated dolbeault complex yields a resolution of the structure sheaf hol x our construction is based on intrinsic and quite explicit semi global koppelman formulas

kink dynamics in the highly discrete sine gordon system abstract we study kink dynamics in a very discrete sine gordon system where the kink width is of the order of the lattice spacing numerical simulations exhibit new properties of kinks in this case they lose the memory of their initial velocity and propagate preferentially at well defined velocities which correspond to quasi steady states while a kink moving at other velocities suffers relatively high rates of radiation of small amplitude oscillations when a small external driving force is applied to the system the same velocities appear as plateus in the strongly nonlinear mobility of the kink the energy radiated by the kink is calculated for a simple model that preserves the discrete character of the system and the preferential velocities for the kink are obtained to good accuracy similar results may be expected to be valid for other discrete systems manifesting topological solitons the numerical simulations reveal also new stable multiple kink excitations which can propagate almost freely in extremely discrete systems where ordinary simple kinks are pinned to the lattice by discreteness the stability of the multiple kinks is discussed

chaos transition of soliton motion on a microstructured lattice abstract soliton motion on a damped driven one dimensional lattice is proposed the lattice model is made of a one dimensional chain equipped at each point with a rotatory molecule the set of coupled difference differential equations for the molecule rotation is obtained for the discrete system using a semi discrete kink solution for which its center of mass is considered as a collective variable we derive the equation of motion for the kink location we show that the kink location is then governed by the equation of the damped sinusoidally driven pendulum when the first order corrections are dropped by means of numerical simulations carried out directly on the lattice model the complex dynamics of the kink location is investigated and compared to that of the well known forced pendulum for some ranges of the control parameters the numerical simulations exhibit a cascade of odd subharmonic components as well as strange attractors which characterizes the chaotic behavior of the kink motion these results agree fairly well with the main features of the transition to chaos for the damped driven pendulum

a quasi periodic route to chaos in a near integrable pde abstract pattern formation and transitions to chaos are described for the damped ac driven one dimensional periodic sine gordon equation in a nonlinear schrodinger regime a generic quasi periodic route to intermittent chaos is exhibited in detail using a range of dynamical systems diagnostics in addition a nonlinear spectral transform is exploited i to identify and quantify coordinates of space time attractors in terms of a small number of soliton modes of the underlying integrable system ii to use these analytic coordinates to identify homoclinic orbits as possible sources of chaos and iii to demonstrate the significance of energy transfer between coherent and extended states in this chaotic system

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem