two step estimation of functional linear models with applications to longitudinal data functional linear models are useful in longitudinal data analysis they include many classical and recently proposed statistical models for longitudinal data and other functional data recently smoothing spline and kernel methods have been proposed for estimating their coefficient functions nonparametrically but these methods are either intensive in computation or inefficient in performance to overcome these drawbacks in this paper a simple and powerful two step alternative is proposed in particular the implementation of the proposed approach via local polynomial smoothing is discussed methods for estimating standard deviations of estimated coefficient functions are also proposed some asymptotic results for the local polynomial estimators are established two longitudinal data sets one of which involves time dependent covariates are used to demonstrate the approach proposed simulation studies show that our two step approach improves the kernel method proposed by hoover and co workers in several aspects such as accuracy computational time and visual appeal of the estimators

the cr geometry of weighted extremal kahler and sasaki metrics we establish an equivalence between conformally einstein maxwell kahler 4 manifolds recently studied in apostolov et al j reine angew math 721 109 147 2016 apostolov and maschler j eur math soc 21 1319 1360 2019 futaki and ono j math soc jpn 70 1493 1521 2018 koca et al ann glob anal geom 50 29 46 2016 lebrun einstein maxwell equations extremal kahler metrics and seiberg witten theory in the many facets of geometry a tribute to nigel hitchin oxford university press oxford pp 17 33 2009 lebrun j geom phys 91 171 2015 and lebrun commun math phys 344 621 653 2016 and extremal kahler 4 manifolds in the sense of calabi extremal kahler metrics seminar on differential geometry annals of mathematics studies vol 102 pp 259 290 princeton university press princeton 1982 with nowhere vanishing scalar curvature the corresponding pairs of kahler metrics arise as transversal kahler structures of sasaki metrics compatible with the same cr structure and having commuting sasaki reeb vector fields this correspondence extends to higher dimensions using the notion of a weighted extremal kahler metric apostolov et al in levi kahler reduction of cr structures product of spheres and toric geometry arxiv 1708 05253 apostolov et al in weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 lahdili in j geom anal 29 542 568 2019 lahdili in int math res not arxiv 1710 00235 lahdili in proc lond math soc 119 1065 1114 2019 illuminating and uniting several explicit constructions in kahler and sasaki geometry it also leads to new existence and non existence results for extremal sasaki metrics suggesting a link between the notions of relative weighted k stability of a polarized variety introduced in apostolov et al weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 and lahdili proc lond math soc 119 1065 1114 2019 and relative k stability of the kahler cone corresponding to a sasaki polarization studied in boyer and van coevering math res lett 25 1 19 2018 and collins and szekelyhidi j differ geom 109 81 109 2018

the map between conformal hypercomplex hyper kahler and quaternionic kahler geometry we review the general properties of target spaces of hypermultiplets which are quaternionic like manifolds and discuss the relations between these manifolds and their symmetry generators we explicitly construct a one to one map between conformal hypercomplex manifolds i e those that have a closed homothetic killing vector and quaternionic manifolds of one quaternionic dimension less an important role is played by transformations relating complex structures on conformal hypercomplex manifolds and connections on quaternionic manifolds in this map the subclass of conformal hyper kahler manifolds is mapped to quaternionic kahler manifolds we relate the curvatures of the corresponding manifolds and furthermore map the symmetries of these manifolds to each other

optimizability and estimatability for infinite dimensional linear systems an infinite dimensional linear system described by dot x t ax t bu t t geq 0 is said to be optimizable if for every initial state x 0 an input u in l 2 can be found such that x in l 2 here a is the generator of a strongly continuous semigroup on a hilbert space and b is an admissible control operator for this semigroup in this paper we investigate optimizability also known as the finite cost condition and its dual estimatability we explore the connections with stabilizability and detectability we give a very general theorem about the equivalence of input output stability and exponential stability of well posed linear systems the two are equivalent if the system is optimizable and estimatable we conclude that a well posed system is exponentially stable if and only if it is dynamically stabilizable and input output stable we illustrate the theory by two examples based on pdes in two or more space dimensions the wave equation and a structural acoustics model

truncated linear models for functional data summary a conventional linear model for functional data involves expressing a response variable y in terms of the explanatory function x t via the model where a is a scalar b is an unknown function and is a compact interval however in some problems the support of b or x say is a proper and unknown subset of and is a quantity of particular practical interest motivated by a real data example involving particulate emissions we develop methods for estimating we give particular emphasis to the case where 0 and suggest two methods for estimating a b and jointly we introduce techniques for selecting tuning parameters and we explore properties of our methodology by using both simulation and the real data example mentioned above additionally we derive theoretical properties of the methodology and discuss implications of the theory our theoretical arguments give particular emphasis to the problem of identifiability

weak exponential stability of linear time varying differential behaviors abstract we develop a new approach to exponential stability of linear time varying ltv differential behaviors that is analogous to that in our paper on exponential stability of discrete ltv behaviors h bourles et al 2015 5 stability theory for differential state space systems with smooth coefficients is an important subject in the literature for differential ltv behaviors with arbitrary smooth coefficients there is no reasonable stability theory therefore we restrict the smooth varying coefficients to functions that are defined by means of locally convergent puiseux series all rational functions are of this type we introduce a new kind of behaviors and prove a module behavior duality for these we define a new notion of weak exponential stability w e s of a behavior b and its associated finitely generated f g module m and show that the w e s modules and behaviors are closed under isomorphisms subobjects factor objects and extensions the standard uniform exponential stability of state space equations is not preserved under behavior isomorphisms and unsuitable for a behavioral theory in the main result we assume a nonzero f g torsion module m and its associated autonomous behavior b such a module may be regular or irregular singular according to the galois theory of differential equations if it is nonzero and regular singular it is never w e s for irregular singular m we characterize w e s of most b algebraically and constructively

smoothing splines estimators for functional linear regression the paper considers functional linear regression where scalar responses y1 yn are modeled in dependence of random functions x1 xn we propose a smoothing splines estimator for the functional slope parameter based on a slight modification of the usual penalty theoretical analysis concentrates on the error in an out ofsample prediction of the response for a new random function xn 1 it is shown that rates of convergence of the prediction error depend on the smoothness of the slope function and on the structure of the predictors we then prove that these rates are optimal in the sense that they are minimax over large classes of possible slope functions and distributions of the predictive curves for the case of models with errors in variables the smoothing spline estimator is modified by using a denoising correction of the covariance matrix of discretized curves the methodology is then applied to a real case study where the aim is to predict the maximum of the concentration of ozone by using the curve of this concentration measured the preceding day

on properties of functional principal components analysis summary functional data analysis is intrinsically infinite dimensional functional principal component analysis reduces dimension to a finite level and points to the most significant components of the data however although this technique is often discussed its properties are not as well understood as they might be we show how the properties of functional principal component analysis can be elucidated through stochastic expansions and related results our approach quantifies the errors that arise through statistical approximation in successive terms of orders n 1 2 n 1 n 3 2 where n denotes sample size the expansions show how spacings among eigenvalues impact on statistical performance the term of size n 1 2 illustrates first order properties and leads directly to limit theory which describes the dominant effect of spacings thus for example spacings are seen to have an immediate first order effect on properties of eigenfunction estimators but only a second order effect on eigenvalue estimators our results can be used to explore properties of existing methods and also to suggest new techniques in particular we suggest bootstrap methods for constructing simultaneous confidence regions for an infinite number of eigenvalues and also for individual eigenvalues and eigenvectors

toric self dual einstein metrics as quotients we use the quaternion kahler reduction technique to study old and new self dual einstein metrics of negative scalar curvature with at least a two dimensional isometry group and relate the quotient construction to the hyperbolic eigenfunction ansatz we focus in particular on the semi quaternion kahler quotients of semi quaternion kahler hyperboloids analysing the completeness and topology and relating them to the self dual einstein hermitian metrics of apostolov gauduchon and bryant

adaptive functional linear regression we consider the estimation of the slope function in functional linear regression where scalar responses are modeled in dependence of random functions cardot and johannes j multivariate anal 101 2010 395 408 have shown that a thresholded projection estimator can attain up to a constant minimax rates of convergence in a general framework which allows us to cover the prediction problem with respect to the mean squared prediction error as well as the estimation of the slope function and its derivatives this estimation procedure however requires an optimal choice of a tuning parameter with regard to certain characteristics of the slope function and the covariance operator associated with the functional regressor as this information is usually inaccessible in practice we investigate a fully data driven choice of the tuning parameter which combines model selection and lepski s method it is inspired by the recent work of goldenshluger and lepski ann statist 39 2011 1608 1632 the tuning parameter is selected as minimizer of a stochastic penalized contrast function imitating lepski s method among a random collection of admissible values this choice of the tuning parameter depends only on the data and we show that within the general framework the resulting data driven thresholded projection estimator can attain minimaxrates up to a constant over a variety of classes of slope functions and covariance operators the results are illustrated considering different configurations which cover in particular the prediction problem as well as the estimation of the slope and its derivatives a simulation study shows the reasonable performance of the fully data driven estimation procedure

twists versus modifications the twist construction is a geometric t duality that produces new manifolds from old works well with for example hypercomplex structures and is easily inverted it tends to destroy properties such as the hyperkahler condition on the other hand modifications preserve the hyperkahler property but do not have an obvious inversion in this paper we show how elementary deformations provide a link between the two constructions and use the twist construction to build hyperkahler and strong hkt structures in the process we provide a full classification of complete hyperkahler four manifolds with tri hamiltonian symmetry and study a number singular phenomena in detail

methodology and convergence rates for functional linear regression in functional linear regression the slope parameter is a function therefore in a nonparametric context it is determined by an infinite number of unknowns its estimation involves solving an ill posed problem and has points of contact with a range of methodologies including statistical smoothing and deconvolution the standard approach to estimating the slope function is based explicitly on functional principal components analysis and consequently on spectral decomposition in terms of eigenvalues and eigen functions we discuss this approach in detail and show that in certain circumstances optimal convergence rates are achieved by the pca technique an alternative approach based on quadratic regularisation is suggested and shown to have advantages from some points of view

stabilizability of linear time varying and uncertain linear systems feedback stabilization of linear time varying and uncertain linear systems is considered it is proved that given a stabilizing dynamic linear state feedback controller one can always construct a stabilizing nondynamic linear state feedback controller a similar result is shown for uncertain linear systems a linear time varying system can be stabilized by dynamic output feedback if and only if it admits a coprime factorization

stabilising compensators for linear time varying differential systems abstractin this paper we describe a constructive test to decide whether a given linear time varying ltv differential system admits a stabilising compensator for the control tasks of tracking disturbance rejection or model matching and construct and parametrise all of them if at least one exists in analogy to the linear time invariant lti case the ring of stable rational functions noncommutative in the ltv situation and the kucera youla parametrisation play prominent parts in the theory we transfer blumthaler s thesis from the lti to the ltv case and sharpen complete and simplify the corresponding results in the book linear time varying systems by bourles and marinescu

a reproducing kernel hilbert space approach to functional linear regression we study in this paper a smoothness regularization method for functional linear regression and provide a unified treatment for both the prediction and estimation problems by developing a tool on simultaneous diagonalization of two positive definite kernels we obtain shaper results on the minimax rates of convergence and show that smoothness regularized estimators achieve the optimal rates of convergence for both prediction and estimation under conditions weaker than those for the functional principal components based methods developed in the literature despite the generality of the method of regularization we show that the procedure is easily implementable numerical results are obtained to illustrate the merits of the method and to demonstrate the theoretical developments

homogeneous almost quaternion hermitian manifolds an almost quaternion hermitian structure on a riemannian manifold m 4n g is a reduction of the structure group of m to mathrm sp n mathrm sp 1 subset text so 4n in this paper we show that a compact simply connected homogeneous almost quaternion hermitian manifold of non vanishing euler characteristic is either a wolf space or mathbb s 2 times mathbb s 2 or the complex quadric text so 7 mathrm u 3

normalized coprime factorizations for linear time varying systems abstract in this paper we show that a finite dimensional linear time varying continuous time system admits normalized coprime factorizations if and only if it admits a stabilizable and detectable realization we construct state space formulas for these factorizations using the stabilizing solutions to standard riccati differential equations in the process we give a simple proof that stabilizability and detectability are sufficient to ensure the existence of such solutions based on these results and on recent advances in the theory of h optimization we present an algorithm to compute the distance between two systems in the gap metric

on asymptotic stability of linear time varying systems this paper is concerned with asymptotic stability analysis of linear time varying ltv systems with the help of the notion of stable functions some differential lyapunov inequalities dlis based necessary and sufficient conditions are derived for testing asymptotic stability exponential stability and uniformly exponential stability of general ltv systems with the help of the concept of non uniformly exponential stability a class of upper triangular ltv systems is carefully investigated based on the developed stability analysis approaches a couple of numerical examples with some of them borrowed from the literature is provided to illustrate the effectiveness of the proposed theoretical results

functional linear regression analysis for longitudinal data we propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data where both the predictor and response are functions of a covariate such as time predictor and response processes have smooth random trajectories and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects in longitudinal studies the number of repeated measurements per subject is often small and may be modeled as a discrete random number and accordingly only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit we propose a functional regression approach for this situation using functional principal component analysis where we estimate the functional principal component scores through conditional expectations this allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory the resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories asymptotic properties for a sample of n subjects are investigated under mild conditions as n oo and we obtain consistent estimation for the regression function besides convergence results for the components of functional linear regression such as the regression parameter function we construct asymptotic pointwise confidence bands for the predicted trajectories a functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced extending the standard r 2 to the functional case the proposed methods are illustrated with a simulation study longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index

detectability and stabilizability of time varying discrete time linear systems the concepts of detectability and stabilizability are explored for time varying systems we study duality invariance under feedback an extended version of the lemma of lyapunov existence of stabilizing feedback laws linear quadratic filtering and control and the existence of approximate canonical forms

stability analysis of linear time varying time delay systems by non quadratic lyapunov functions with indefinite derivatives abstract this paper presents a stability analysis of linear time varying ltv time delay systems by using non quadratic lyapunov functions and functionals two types of sufficient conditions are proposed for testing several different kinds of stability such as asymptotic stability exponential stability and uniformly exponential stability for ltv systems without delay then by constructing suitable non quadratic lyapunov functions functionals that are respectively the time varying weighted l 1 and l norms of the state variables and by using properties of the uniformly asymptotically stable uas function and the recently established improved razumikhin and krasovskii stability theorems both delay dependent and delay independent conditions are proposed for testing uniformly exponential stability of a class of ltv time delay systems the time derivatives of the non quadratic lyapunov functions functionals along the solutions are allowed to be indefinite namely to take both negative and positive values numerical examples show that the non quadratic lyapunov functions functionals based methods are more efficient than the existing ones that are based on quadratic functions functionals

supersymmetric quantum theory and non commutative geometry classical differential geometry can be encoded in spectral data such as connes spectral triples involving supersymmetry algebras in this paper we formulate non commutative geometry in terms of supersymmetric spectral data this leads to generalizations of connes non commutative spin geometry encompassing non commutative riemannian symplectic complex hermitian and hyper kahler geometry a general framework for non commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples in particular the non commutative torus and the non commutative 3 sphere are studied in some detail

multi moment maps we introduce a notion of moment map adapted to actions of lie groups that preserve a closed three form we show existence of our multi moment maps in many circumstances including mild topological assumptions on the underlying manifold such maps are also shown to exist for all groups whose second and third lie algebra betti numbers are zero we show that these form a special class of solvable lie groups and provide a structural characterisation we provide many examples of multi moment maps for different geometries and use them to describe manifolds with holonomy contained in g2 preserved by a two torus symmetry in terms of tri symplectic geometry of four manifolds

clt in functional linear regression models we propose in this work to derive a clt in the functional linear regression model the main difficulty is due to the fact that estimation of the functional parameter leads to a kind of ill posed inverse problem we consider estimators that belong to a large class of regularizing methods and we first show that contrary to the multivariate case it is not possible to state a clt in the topology of the considered functional space however we show that we can get a clt for the weak topology under mild hypotheses and in particular without assuming any strong assumptions on the decay of the eigenvalues of the covariance operator rates of convergence depend on the smoothness of the functional coefficient and on the point in which the prediction is made

stabilizability of linear time varying systems abstract for linear time varying systems with bounded system matrices we discuss the problem of stabilizability by linear state feedback for example it is shown that complete controllability implies the existence of a feedback so that the closed loop system is asymptotically stable we also show that the system is completely controllable if and only if the lyapunov exponent is arbitrarily assignable by a suitable feedback for uniform exponential stabilizability and the assignability of the bohl exponent this property is known also dynamic feedback does not provide more freedom to address the stabilization problem the unifying tools for our results are two finite l 2 cost conditions the distinction of exponential and uniform exponential stabilizability is then a question of whether the finite cost condition is uniform in the initial time or not

geometry of hyper kahler connections with torsion the internal space of a n 4 supersymmetric model with wess zumino term has a connection with totally skew symmetric torsion and holonomy in sp n we study the mathematical background of this type of connection in particular we relate it to classical hermitian geometry construct homogeneous as well as inhomogeneous examples characterize it in terms of holomorphic data develop its potential theory and reduction theory

rigidity for multi taub nut metrics this paper provides a classification result for gravitational instantons with cubic volume growth and cyclic fundamental group at infinity it proves that a complete hyperkahler manifold asymptotic to a circle fibration over the euclidean three space is either the standard r 3 s 1 or a multi taub nut manifold in par ticular the underlying complex manifold is either c c z or a minimal resolution of a cyclic kleinian singularity

painleve formulas of the limiting distributions for nonnull complex sample covariance matrices in a recent study of large non null sample covariance matrices a new sequence of functions gener alizing the gue tracy widom distribution of random matrix theory was obtained this paper derives painleve formulas of these functions and use them to prove that they are indeed distribution functions applications of these new distribution functions to last passage percolation queues in tandem and to tally asymmetric simple exclusion process are also discussed as a part of the proof a representation of orthogonal polynomials on the unit circle in terms of an operator on a discrete set is presented

minimax and adaptive prediction for functional linear regression this article considers minimax and adaptive prediction with functional predictors in the framework of functional linear model and reproducing kernel hilbert space minimax rate of convergence for the excess prediction risk is established it is shown that the optimal rate is determined jointly by the reproducing kernel and the covariance kernel in particular the alignment of these two kernels can significantly affect the difficulty of the prediction problem in contrast the existing literature has so far focused only on the setting where the two kernels are nearly perfectly aligned this motivates us to propose an easily implementable data driven roughness regularization predictor that is shown to attain the optimal rate of convergence adaptively without the need of knowing the covariance kernel simulation studies are carried out to illustrate the merits of the adaptive predictor and to demonstrate the theoretical results

new hkt manifolds arising from quaternionic representations we give a procedure for constructing an 8n dimensional hkt lie algebra starting from a 4n dimensional one by using a quaternionic representation of the latter the strong respectively weak hyper kahler balanced condition is preserved by our construction as an application of our results we obtain a new compact hkt manifold with holonomy in sl n mathbb h which is not a nilmanifold we find in addition new compact strong hkt manifolds we also show that every kahler lie algebra equipped with a flat torsion free complex connection gives rise to an hkt lie algebra we apply this method to two distinguished 4 dimensional kahler lie algebras thereby obtaining two conformally balanced hkt metrics in dimension 8 both techniques prove to be an effective tool for giving the explicit expression of the corresponding hkt metrics

complete ricci flat kahler manifolds of infinite topological type we display an infinite dimensional family of complete ricci flat kahler manifolds of complex dimension 2 for which the second homology is infinitely generated these are obtained from the gibbons hawking ansatz 2 by using infinitely many sparsely distributed centers

existence of hkt metrics on hypercomplex manifolds of real dimension 8 abstract a hypercomplex manifold m is a manifold equipped with three complex structures i j k satisfying quaternionic relations such a manifold admits a canonical torsion free connection preserving the quaternion action called the obata connection a quaternionic hermitian metric is a riemannian metric which is invariant with respect to unitary quaternions such a metric is called hyperkahler with torsion hkt for short if it is locally obtained as the hessian of a function averaged with quaternions an hkt metric is a natural analogue of a kahler metric on a complex manifold we push this analogy further proving a quaternionic analogue of the result of buchdahl and of lamari that a compact complex surface m admits a kahler structure if and only if b 1 m is even we show that a hypercomplex manifold m with the obata holonomy contained in s l 2 h admits an hkt structure if and only if h 1 o m i is even dimensional

statistical inferences for functional data with modem technology development functional data are being observed frequently in many scientific fields a popular method for analyzing such functional data is smoothing first then estimation that is statistical inference such as estimation and hypothesis testing about functional data is conducted based on the substitution of the underlying individual functions by their reconstructions obtained by one smoothing technique or another however little is known about this substitution effect on functional data analysis in this paper this problem is investigated when the local polynomial kernel lpk smoothing technique is used for individual function reconstructions we find that under some mild conditions the substitution effect can be ignored asymptotically based on this we construct lpk reconstruction based estimators for the mean covariance and noise variance functions of a functional data set and derive their asymptotics we also propose a gcv rule for selecting good bandwidths for the lpk reconstructions when the mean function also depends on some time independent covariates we consider a functional linear model where the mean function is linearly related to the covariates but the covariate effects are functions of time the lpk reconstruction based estimators for the covariate effects and the covariance function are also constructed and their asymptotics are derived moreover we propose a l 2 norm based global test statistic for a general hypothesis testing problem about the covariate effects and derive its asymptotic random expression the effect of the bandwidths selected by the proposed gcv rule on the accuracy of the lpk reconstructions and the mean function estimator is investigated via a simulation study the proposed methodologies are illustrated via an application to a real functional data set collected in climatology

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

a uniform estimate for general quaternionic calabi problem with appendix by daniel barlet abstract we prove a c 0 a priori estimate on a solution of the quaternionic calabi problem on an arbitrary compact connected hkt manifold this generalizes earlier results 9 and 7 where this result was proven under certain extra assumptions on the manifold

functional linear regression that s interpretable regression models to relate a scalar y to a functional predictor x t are becoming increasingly common work in this area has concentrated on estimating a coefficient function t with y related to x t through f t x t dt regions where t 0 correspond to places where there is a relationship between x t and y alternatively points where t 0 indicate no relationship hence for interpretation purposes it is desirable for a regression procedure to be capable of producing estimates of t that are exactly zero over regions with no apparent relationship and have simple structures over the remaining regions unfortunately most fitting procedures result in an estimate for t that is rarely exactly zero and has unnatural wiggles making the curve hard to interpret in this article we introduce a new approach which uses variable selection ideas applied to various derivatives of t to produce estimates that are both interpretable flexible and accurate we call our method functional linear regression that s interpretable flirti and demonstrate it on simulated and real world data sets in addition non asymptotic theoretical bounds on the estimation error are presented the bounds provide strong theoretical motivation for our approach

proportional local assignability of lyapunov spectrum of linear discrete time varying systems we consider a local version of the pole assignment problem for linear discrete time varying systems our aim is to obtain sufficient conditions to place the lyapunov spectrum of the closed loop sys

twist geometry of the c map we discuss the geometry of the c map from projective special kahler to quaternionic kahler manifolds using the twist construction to provide a global approach to hitchin s description as found by alexandrov et al and alekseevsky et al this is related to the quaternionic flip of haydys we prove uniqueness statements for several steps of the construction in particular we show that given a hyperkahler manifold with a rotating symmetry there is essentially only a one parameter degree of freedom in constructing a quaternionic kahler manifold of the same dimension we demonstrate how examples on group manifolds arise from this picture