behaviour of the output error identification algorithm for small stepsize gains in the output error identification algorithm it is possible to adjust a gain parameter to trade off for example sensitivity to noise and capacity to track plant parameter variations the rate of convergence of the ideal algorithm obtained in the noise free time invariant plant parameter case is related to the magnitude of this gain parameter the analysis applies to a broader class of adaptive problems than output error identification and in particular a positive real transfer function appearing in the problem formulation is allowed to be nonrational an alternative to the usual lyapunov based analysis is then needed

atiyah class and sheaf counting on local calabi yau fourfolds abstract we discuss donaldson thomas dt invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non compact calabi yau fourfold given by the total space of a rank 2 bundle on the surface we prove that in certain cases when the rank 2 bundle is chosen appropriately the universal truncated atiyah class of these codimension 2 sheaves reduces to one defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor threefold embedded in the ambient fourfold such reduction property of universal atiyah class enables us to relate our fourfold dt theory to a reduced dt theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface using results in 15 16 we finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic k3

on the homology of the hilbert scheme of points in the plane geir ellingsrud 1 and stein arild stromme 2 i matematisk institutt universitetet i oslo blindern n oslo 3 norway 2 matematisk institutt universitetet i bergen n 5014 bergen norway although several authors have been interested in the hilbert scheme hilba ip z parametrizing finite subschemes of length d in the projective plane i1 i2 f 1 f2 br among others not much is known about the topological properties of this space the picard group has been calculated i f2 and the homology groups of hilb3 ip 2 have been computed hi in this paper we give a precise description of the additive structure of the homology of hilba ip2 applying the results of birula bialynicki b1 b2 on the cellular decompositions de fined by a torus action to the natural action of a maximal torus of sl 3 on hilbe ipz a rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the chow groups and the homology groups are isomorphisms in particular there is no odd homology and the homology groups are all free the main objective of this work is to compute their ranks the betti numbers of hilba ipz as a byproduct of our method we get similar results on the homology of the punctual hilbert scheme and of the hilbert scheme of points in the affine plane it seems natural to generalize our results to any toric smooth surface however we give the results only for the rational ruled surfaces if with an indication of the necessary changes in the proofs for simplicity we work over the field of complex numbers but with an appropriate interpretation of the word homology our results remain valid over any base field

on differentiable vectors for representations of infinite dimensional lie groups in this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations g gl v of an infinite dimensional lie group g on a locally convex space v the first class of results concerns the space v of smooth vectors if g is a banach lie group we define a topology on the space v of smooth vectors for which the action of g on this space is smooth if v is a banach space then v is a frechet space this applies in particular to c dynamical systems a g where g is a banach lie group for unitary representations we show that a vector v is smooth if the corresponding positive definite function g v v is smooth the second class of results concerns criteria for ck vectors in terms of operators of the derived representation for a banach lie group g acting on a banach space v in particular we provide for each k n examples of continuous unitary representations for which the space of ck 1 vectors is trivial and the space of ck vectors is dense

differentiable vectors and unitary representations of frechet lie supergroups a locally convex lie group g has the trotter property if for every x 1 x 2 in mathfrak g begin aligned exp g t x 1 x 2 lim n rightarrow infty left exp g left frac t n x 1 right exp g left frac t n x 2 right right n end aligned holds uniformly on compact subsets of mathbb r all locally exponential lie groups have this property but also groups of automorphisms of principal bundles over compact smooth manifolds a key result of the present article is that if g has the trotter property pi g rightarrow mathrm gl v is a continuous representation of g on a locally convex space and v in v is a vector such that overline mathtt d pi x v frac d dt t 0 pi exp g tx v exists for every x in mathfrak g then the map mathfrak g rightarrow v x mapsto overline mathtt d pi x v is linear using this result we conclude that for a representation of a locally exponential frechet lie group g on a metrizable locally convex space the space of mathcal c k vectors coincides with the common domain of the k fold products of the operators overline mathtt d pi x for unitary representations on hilbert spaces the assumption of local exponentiality can be weakened to the trotter property as an application we show that for smooth resp analytic unitary representations of frechet lie supergroups g mathfrak g where g has the trotter property the common domain of the operators of mathfrak g mathfrak g overline 0 oplus mathfrak g overline 1 can always be extended to the space of smooth resp analytic vectors for g

persistence of excitation conditions and the convergence of adaptive schemes in the study of the behavior of adaptive filtering algorithms persistence of excitation of the input process arises as a sufficient condition for convergence and perhaps more importantly for convergence rate of the parameter estimates in this paper the underlying nature of the persistence requirement is presented and discussed relating its normal specification in terms of moment conditions with covariance decays etc to sample path properties deterministic and stochastic persistence conditions and persistence measures are treated as well as persistence conditions for output error equation error and adaptive control schemes

integrating representations of banach lie algebras abstract we give two integrability criteria for representations of banach lie algebras as skew symmetric unbounded operators on a dense domain of a hilbert space one of them is based on analytic vectors

deformation obstruction theory for complexes via atiyah and kodaira spencer classes we give a universal approach to the deformation obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family we recover and generalise the obstruction class of lowen and lieblich and prove that it is a product of atiyah and kodaira spencer classes this allows us to obtain deformation invariant virtual cycles on moduli spaces of objects of the derived category on threefolds

analysis of the normalized lms algorithm with gaussian inputs the lms adaptive filter algorithm requires a priori knowledge of the input power level to select the algorithm gain parameter for stability and convergence since the input power level is usually one of the statistical unknowns it is normally estimated from the data prior to beginning the adaptation process it is then assumed that the estimate is perfect in any subsequent analysis of the lms algorithm behavior in this paper the effects of the power level estimate are incorporated in a data dependent that appears explicitly within the algorithm the transient mean and second moment behavior of the modified lms nlms algorithm are evaluated taking into account the explicit statistical dependence of upon the input data the mean behavior of the algorithm is shown to converge to the wiener weight a constant coefficient matrix difference equation is derived for the weight fluctuations about the wiener weight the equation is solved for a white data covariance matrix and for the adaptive line enhancer with a single frequency input in steady state for small expressions for the misadjustment error are also presented it is shown for the white data covariance matrix case that the averaging of about ten data samples causes negligible degradation as compared to the lms algorithm in the ale application the steady state weight fluctuations are shown to be mode dependent being largest at the frequency of the input

convergence analysis of adaptive linear estimation for dependent stationary processes the convergence properties of an adaptive linear mean square estimator that uses a modified lms algorithm are established for generally dependent processes bounds on the mean square error of the estimates of the filter coefficients and on the excess error of the estimate of the signal are derived for input processes which are either strong mixing or asymptotically uncorrelated it is shown that the mean square deviation is bounded by a constant multiple of the adaptation step size and that the same holds for the excess error of the signal estimation the present findings extend earlier results in the literature obtained for independent and m dependent input data

behavior of the e normalized lms algorithm with gaussian inputs the first and second moment behavior of one variation of the normalized lms e nlms algorithm is investigated for a white covariance matrix and gaussian statistics for the data for this model it is shown that the e nlms algorithm has neither behavior independent of the input data power nor a performance significantly better than the lms algorithm for which the input power level also must be known a priori hence based upon the results of the analysis it is recommended that the algorithm not be used in place of the lms for known input power levels or in place of the nlms for unknown input power levels

asymptotic behavior of an adaptive estimation algorithm with application to m dependent data theoretical results are presented concerning the asymptotic behavior of the parameter estimates generated by an adaptive algorithm in stationary dependent random situations proofs are exhibited in the general case and derived for the case of statistical dependence in the input for a finite number of lags it is found that the estimate mean error norm converges to an asymptotic bound giving a finite bias generally nonzero and that the asymptotic mean norm square error is bounded and can be arbitrarily reduced by decreasing the adaptation factor

euler characteristics of su 2 instanton moduli spaces on rational elliptic surfaces recently minahan nemeschansky vafa and warner computed partition functions for n 4 topological yang mills theory on rational elliptic surfaces in particular they computed generating functions of euler characteristics of su 2 instanton moduli spaces in mathematics they are expected to coincide with those of gieseker compactifications in this paper we compute euler characteristics of these spaces and show that our results coincide with theirs we also check the modular property of z su 2 and z so 3 conjectured by vafa and witten

theta functions and hodge numbers of moduli spaces of sheaves on rational surfaces we compute generating functions for the hodge numbers of the moduli spaces of h stable rank 2 sheaves on a rational surface s in terms of theta functions for indefinite lattices if h lies in the closure of the ample cone and has self intersection 0 it follows that the generating functions are jacobi forms in particular the generating functions for the euler numbers can be expressed in terms of modular forms and their transformation behaviour is compatible with the predictions of s duality we also express the generating functions for the signatures in terms of modular forms it turns out that these generating functions are also with respect to another developing parameter the generating function for the donaldson invariants of s evaluated on all powers of the point class

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one