nonabelian bundle gerbes their differential geometry and gauge theory bundle gerbes are a higher version of line bundles we present nonabelian bundle gerbes as a higher version of principal bundles connection curving curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates these are the gauge fields needed for the construction of yang mills theories with 2 form gauge potential

lie bialgebroids and poisson groupoids lie bialgebras arise as infinitesimal invariants of poisson lie groups a lie bialgebra is a lie algebra g with a lie algebra structure on the dual g which is compatible with the lie algebra g in a certain sense for a poisson group g the multiplicative poisson structure induces a lie algebra structure on the lie algebra dual g which makes g g into a lie bialgebra in fact there is a one one correspondence between poisson lie groups and lie bialgebras if the lie groups are assumed to be simply connected 7 16 19 the importance of poisson lie groups themselves arises in part from their role as classical limits of quantum groups 8 and in part because they provide a class of poisson structures for which the realization problem is tractable 15 poisson groupoids were introduced by weinstein 24 as a generalization of both poisson lie groups and the symplectic groupoids which arise in the integration of arbitrary poisson manifolds 4 11 he noted that the lie algebroid dual a g of a poisson groupoid g itself has a lie algebroid structure but did not develop the infinitesimal structure further in this paper we introduce and study a natural infinitesimal invariant for poisson groupoids the lie bialgebroids of the title our ultimate purpose is to develop a lie theory for poisson groupoids parallel to that for poisson groups in this paper we are primarily concerned with the first half of this process that is with the construction of the lie bialgebroid of a poisson groupoid after the preliminary 2 in which we describe the generalization to arbitrary lie algebroids of the exterior calculus and schouten calculus in 3 we define a lie bialgebroid to be a lie algebroid a whose dual a is also equipped with a lie algebroid structure such that the coboundary operator d a a associated to a satisfies a cocycle equation with respect to a the lie algebra of sections of a this is clearly a straightforward extension of the concept of a lie bialgebra 16 but cannot be formalized in lie algebroid cohomological terms since there is no satisfactory adjoint representation for a general lie algebroid most of 3 is devoted to proving that this definition is self dual if a a is a lie bialgebroid then a a is also in 4 we briefly consider the special case of lie bialgebroids satisfying a triangularity condition which include some important examples such as the usual triangular lie bialgebras and lie bialgebroids associated to poisson manifolds the techniques used in 2 4 are similar to those known for lie bialgebras it would be possible by suitably generalizing the proof for poisson groups to prove

higher gauge theory differential versus integral formulation the term higher gauge theory refers to the generalization of gauge theory to a theory of connections at two levels essentially given by 1 and 2 forms so far there have been two approaches to this subject the differential picture uses non abelian 1 and 2 forms in order to generalize the connection 1 form of a conventional gauge theory to the next level the integral picture makes use of curves and surfaces labeled with elements of non abelian groups and generalizes the formulation of gauge theory in terms of parallel transports we recall how to circumvent the classic no go theorems in order to define non abelian surface ordered products in the integral picture we then derive the differential picture from the integral formulation under the assumption that the curve and surface labels depend smoothly on the position of the curves and surfaces we show that some aspects of the no go theorems are still present in the differential but not in the integral picture this implies a substantial structural

simultaneous confidence intervals for comparing binomial parameters summary to compare proportions with several independent binomial samples we recommend a method of constructing simultaneous confidence intervals that uses the studentized range distribution with a score statistic it applies to a variety of measures including the difference of proportions odds ratio and relative risk for the odds ratio a simulation study suggests that the method has coverage probability closer to the nominal value than ad hoc approaches such as the bonferroni implementation of wald or exact small sample pairwise intervals it performs well even for the problematic but practically common case in which the binomial parameters are relatively small for the difference of proportions the proposed method has performance comparable to a method proposed by piegorsch 1991 biometrics 47 45 52

geometry of maurer cartan elements on complex manifolds the semi classical data attached to stacks of algebroids in the sense of kashiwara and kontsevich are maurer cartan elements on complex manifolds which we call extended poisson structures as they generalize holomorphic poisson structures a canonical lie algebroid is associated to each maurer cartan element we study the geometry underlying these maurer cartan elements in the light of lie algebroid theory in particular we extend lichnerowicz poisson cohomology and koszul brylinski homology to the realm of extended poisson manifolds we establish a sufficient criterion for these to be finite dimensional we describe how homology and cohomology are related through the evens lu weinstein duality module and we describe a duality on koszul brylinski homology which generalizes the serre duality of dolbeault cohomology

existence of lipschitz and semiconcave control lyapunov functions given a locally lipschitz control system which is globally asymptotically controllable at the origin we construct a control lyapunov function for the system which is lipschitz on bounded sets and we deduce the existence of another one which is semiconcave and so locally lipschitz outside the origin the proof relies on value functions and nonsmooth calculus

on brockett s condition for smooth stabilizability and its necessity in a context of nonsmooth feedback the necessity of brockett s condition for stabilizability of nonlinear systems by smooth feedback is shown by an argument based on properties of a degree for set valued maps to persist when the class of controls is enlarged to include discontinuous feedback

simultaneous confidence bounds for relative risks in multiple comparisons to control we discuss the construction of asymptotic simultaneous upper confidence limits that jointly bound relative risks formed by comparing several treatments to a control motivated by a vaccine study we investigate the performance of several methods under such settings inverting the minimum of score statistics together with estimating the correlation matrix of these statistics under the null gives simultaneous coverage rates closest to the nominal level in typical settings of vaccine studies this method proves to be the most powerful of the ones considered but computationally simpler alternatives are also worth exploring when the number of comparisons is large simultaneous lower and two sided confidence intervals are also considered all procedures can be implemented and evaluated using freely available and general r code

global inverse optimal control with guaranteed convergence rates of input affine nonlinear systems in earlier works global inverse optimal controllers were proposed for input affine nonlinear systems with given control lyapunov functions however these controllers do not provide information about the convergence rates if the systems have local homogeneous approximations we can employ homogeneous controllers which locally asymptotically stabilize the origin and specify the convergence rates however homogeneous controllers generally do not attain global stability for nonhomogeneous systems in this paper we design global inverse optimal controllers with guaranteed local convergence rates by utilizing local homogeneity of input affine nonlinear systems if we do not consider inverse optimality we can liberally adjust the sector margins we also clarify that local convergence rates and sector margins are invariant under coordinate transformations

a general notion of global asymptotic controllability for time varying systems and its lyapunov characterization a general notion of global asymptotic controllability to a given equilibrium of a time varying system is introduced and is shown that this property is equivalent to the existence of a lower semicontinuous control lyapunov function the work constitutes an extension of albertini and sontag 1999 paper in same journal dealing with same subject under stricter hypotheses

asymptotic controllability implies feedback stabilization it is shown that every asymptotically controllable system can be globally stabilized by means of some discontinuous feedback law the stabilizing strategy is based on pointwise optimization of a smoothed version of a control lyapunov function iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium a major technical problem and one of the contributions of the present paper concerns the precise meaning of solution when using a discontinuous controller

stabilization with relaxed controls cannot in general be stabilized using a continuous closed loop control u x even if each state separately can be driven asymptotically to the origin an example is analyzed in section 2 in this paper we examine the possibility of stabilizing such systems with a continuous closed loop relaxed control we find indeed that the family of systems stabilizable with relaxed controls is larger than the family of those stabilizable with ordinary controls an even larger class is obtained if the continuity of the closed loop at the origin is not required the latter class includes all one dimensional systems for which states can be driven asymptotically to the origin this result does not hold in two dimensional systems and we provide a counter example it should be pointed that relaxed control type stabilization is used both in theory and in practice the method is known as dither we shall comment on the similarities lyapunov functions for the system 1 help us in the construction of the continuous closed loop stabilizers in fact we find that the existence of a smooth lyapunov function is equivalent to the existence of a stabilizing closed loop which is continuous except possibly at the origin an additional condition on the lyapunov function implies the continuity at the origin as well we present these results in section 4 after a brief introduction of closed loop relaxed controls notations and terminology in section 3 prior to that in section 2 we discuss an example illustrating the power of relaxed controls in the particular case of systems linear in the controls relaxed controls can be replaced by ordinary controls this is discussed in section 5 the role of lyapunov functions in the stability and stabilization theories is of course well known examples of systems with lyapunov functions are available in the literature we display some in section 6 along with general comments on the construction applications and counterexamples including one which cannot be continuously stabilized yet possesses a nonsmooth lyapunov function closed loop stabilization with ordinary controls is analyzed extensively in the literature see sontag 8 sussmann ll and references therein lyapunov functions techniques in stabil

feedback stabilization and lyapunov functions given a locally defined nondifferentiable but lipschitz lyapunov function we employ it in order to construct a discontinuous feedback law which stabilizes the underlying system to any given tolerance a converse result shows that suitable lyapunov functions of this type exist under mild assumptions we also establish that the feedback in question possesses a robustness property relative to measurement error despite the fact that it may not be continuous

p values maximized over a confidence set for the nuisance parameter abstract for testing problems of the form h 0 v v 0 with unknown nuisance parameter various methods are used to deal with the simplest approach is exemplified by the t test where the unknown variance is replaced by the sample variance and the t distribution accounts for estimation of the variance in other problems such as the 2 2 contingency table one conditions on a sufficient statistic for 0 and proceeds as in fisher s exact test because neither of these standard methods is appropriate for all situations this article suggests a new method for handling the unknown this new method is a simple modification of the formal definition of a p value that involves taking a maximum over the nuisance parameter space of a p value obtained for the case when is known the suggested modification is to restrict the maximization to a confidence set for the nuisance parameter after giving a brief justification we give various examples to show how this new method gives improved results for 2 2 tabl

links between asymptotic controllability and persistence of excitation for a class of time varying systems abstract for a class of nonlinear time varying parameterized systems we prove that persistence of excitation implies global asymptotic controllability to the equilibrium with respect to a set of parameters the result is used to derive sufficient conditions for global asymptotic controllability for composite systems

stabilization of nonlinear systems in the plane abstract it is shown that every small time locally controllable system in the plane can locally be asymptotically stabilized by employing locally holder continuous feedback laws as essentially was conjectured by e sontag an explicit algorithm for the construction of such feedback laws is given

duality theorems for etale gerbes on orbifolds abstract let g be a finite group and y a g gerbe over an orbifold b a disconnected orbifold y and a flat u 1 gerbe c on y is canonically constructed from y motivated by a proposal in physics we study a mathematical duality between the geometry of the g gerbe y and the geometry of y twisted by c we prove several results verifying this duality in the contexts of non commutative geometry and symplectic topology in particular we prove that the category of sheaves on y is equivalent to the category of c twisted sheaves on y when y is symplectic we show by a combination of techniques from non commutative geometry and symplectic topology that the chen ruan orbifold cohomology of y is isomorphic to the c twisted orbifold cohomology of y as graded algebras

homogeneous stabilization for input affine homogeneous systems dilated homogeneous systems are local canonical forms of nonlinear control systems for input affine homogeneous systems existence of continuous homogeneous stabilizing controllers is equivalent to existence of differentiable homogeneous control lyapunov functions in this technical note we propose an homogeneous stabilizing controller and an homogeneous inverse optimal controller for multi input affine homogeneous systems with given homogeneous control lyapunov functions

sampled data feedback practical semi global controllability and stabilization for time varying systems we deal with the problems of the closed loop controllability and stabilization for time varying systems whose dynamics are in general unbounded with respect to time it is shown that a general notion of global asymptotic controllability being in general non uniform with respect to initial values of time implies existence of a family of continuous control lyapunov functions and this enables the establishment of practical semi global closed loop controllability by means of a sampled data time varying feedback moreover the usual concept of global asymptotic controllability being uniform with respect to the initial values of time implies practical semi global stabilization by sampled data feedback the work is a continuation of a recent paper of the author in the same journal and extends krichman s 2000 result relying practical stabilization of time varying systems with bounded in time dynamics

groupoid cohomology and extensions we show that haefliger s cohomology for etale groupoids moore s cohomology for locally compact groups and the brauer group of a locally compact groupoid are all particular cases of sheaf or cech cohomology for topological simplicial spaces

planar nonlinear systems practical stabilization and hermes controllability condition abstract we analyze further some results from our recent work 1991 concerning the practical stabilizability problem by means of smooth feedback laws these results are used to establish that if a smooth planar nonlinear system satisfies the hermes controllability condition at its equilibrium then it can be practically stabilized

deformation quantization of gerbes this is the first in a series of articles devoted to deformation quantization of gerbes we introduce basic definitions interpret deformations of a given stack as maurer cartan elements of a differential graded lie algebra dgla and classify deformations of a given gerbe in terms of maurer cartan elements of the dgla of hochschild cochains twisted by the cohomology class of the gerbe we also classify all deformations of a given gerbe on a symplectic manifold as well as provide a deformation theoretic interpretation of the first rozansky witten class

holonomy and parallel transport for abelian gerbes in this paper we establish a one to one correspondence between u 1 gerbes with connections on the one hand and their holonomies for simply connected manifolds or their parallel transports in the general case on the other hand this result is a higher order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds the holonomy of a gerbe with group u 1 on a simply connected manifold m is a group morphism from the thin second homotopy group to u 1 satisfying a smoothness condition where a homotopy between maps from 0 1 2 to m is thin when its derivative is of rank 2 for the non simply connected case holonomy is replaced by a parallel transport functor between two special lie groupoids which we call lie 2 groups the reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case

sufficient lie algebraic conditions for sampled data feedback stabilizability of affine in the control nonlinear systems for general nonlinear autonomous systems a lyapunov characterization for the possibility of semi global asymptotic stabilizability by means of a time varying sampled data feedback is established we exploit this result in order to derive a lie algebraic sufficient condition for sampled data feedback semi global stabilizability of affine in the control nonlinear systems with non zero drift terms the corresponding proposition constitutes an extension of the artstein sontag theorem on feedback stabilization

state constrained feedback stabilization a standard finite dimensional nonlinear control system is considered along with a state constraint set s and a target set sigma it is proven that open loop s constrained controllability to sigma implies closed loop s constrained controllability to the closed delta neighborhood of sigma for any specified delta 0 when the s constrained minimum time function to sigma satisfies a local continuity condition conclusions on closed loop s constrained stabilizability ensue the necessarily discontinuous feedback laws in question are implemented in the sample and hold sense and possess a robustness property with respect to state measurement errors the feedback constructions involve the quadratic infimal convolution of a control lyapunov function with respect to a certain modification of the original dynamics the modified dynamics in effect provide for constraint removal while the convolution operation provides a useful semiconcavity property

remarks on asymptotic controllability and sampled data feedback stabilization for autonomous systems a weak concept of sampled data feedback stabilization for the case of autonomous systems is presented and its lyapunov characterization is established sufficient conditions are derived for the existence of a time varying sampled data feedback exhibiting global stabilization for certain class of nonlinear systems

averaging criteria for asymptotic stability of time varying systems we establish averaging type sufficient conditions for local asymptotic stability for nonlinear time varying systems our main result is based on an extension of the averaging methodology employed in a recent paper by tsinias and stamati published in the same journal dealing with asymptotic stability for slow time varying systems

an extension of artstein s theorem on stabilization by using ordinary feedback integrators abstract we prove that under suitable lyapunov like conditions a single input nonlinear system although it may not be stabilized by an ordinary state feedback law is dynamically stabilize by means of an ordinary feedback integrator

chern weil map for principal bundles over groupoids the theory of principal g bundles over a lie groupoid is an important one unifying various types of principal g bundles including those over manifolds those over orbifolds as well as equivariant principal g bundles in this paper we study differential geometry of these objects including connections and holonomy maps we also introduce a chern weil map for these principal bundles and prove that the characteristic classes obtained coincide with the universal characteristic classes as an application we recover the equivariant chern weil map of bott tu we also obtain an explicit chain map between the weil model and the simplicial model of equivariant cohomology which reduces to the bott shulman map s mathfrak g g to h bg when the manifold is a point

power and sample size for testing homogeneity of relative risks in prospective studies summary power and sample size formulas for testing the homogeneity of relative risks using the score method are presented the homogeneity score test gart 1985 biometrika72 673 677 is formally equivalent to the pearson chi square test although they look different results of this paper may be useful in assessing the validity of the model of a common relative risk before combining several 2 2 tables or in designing a prospective study for detecting heterogeneity of relative risks

test statistics and sample size formulae for comparative binomial trials with null hypothesis of non zero risk difference or non unity relative risk when it is required to establish a materially significant difference between two treatments or alternatively to show that two treatments are equivalent standard test statistics and sample size formulae based on a null hypothesis of no difference no longer apply this paper reviews some of the test statistics and sample size formulae proposed for comparative binomial trials when the null hypothesis is of a specified non zero difference or non unity relative risk methods based on restricted maximum likelihood estimation are recommended and applied to studies of pertussis vaccine

small gain type sufficient conditions for sampled data feedback stabilization for autonomous composite systems small gain type sufficient conditions are established guaranteeing weak global asymptotic stabilization for composite autonomous systems by means of a sampled data feedback our main result generalizes the well known small gain theorem for global asymptotic stability for composite systems with no controls under the presence of the input to state stability the sufficient conditions we propose enable us to derive a set of algebraic sufficient conditions for the existence of a sampled data feedback stabilizer with zero order hold the work is a continuation of author s work published in same journal

approximate interval estimation of the ratio of binomial parameters a review and corrections for skewness various methods for finding confidence intervals for the ratio of binomial parameters are reviewed and evaluated numerically it is found that the method based on likelihood scores koopman 1984 biometrics 40 513 517 miettinen and nurminen 1985 statistics in medicine 4 213 226 performs best in achieving the nominal confidence coefficient but it may distribute the tail probabilities quite disparately using general theory of bartlett 1953 biometrika 40 306 317 1955 biometrika 42 201 203 we correct this method for asymptotic skewness following gart 1985 biometrika 72 673 677 we extend this correction to the case of estimating the common ratio in a series of two by two tables computing algorithms are given and applied to numerical examples parallel methods for the odds ratio and the ratio of poisson parameters are noted

asymptotic controllability and robust asymptotic stabilizability this paper deals with asymptotically controllable systems for which there exists no smooth stabilizing state feedback to investigate the robustness asymptotic stabilization property a new class of hybrid feedbacks with a continuous component and a discrete one is introduced the hybrid patchy feedbacks the notion of solutions is a generalization of pi solutions and euler solutions it is proved that the origin of all globally asymptotically controllable systems can be globally asymptotically stabilized via a hybrid feedback with robustness with respect to measurement noise actuator errors and external disturbances

continuous control lyapunov functions for asymptotically controllable time varying systems this paper shows that for time varying systems global asymptotic controllability to a given closed subset of the state space is equivalent to the existence of a continuous control lyapunov function with respect to the set

non abelian differentiable gerbes abstract we study non abelian differentiable gerbes over stacks using the theory of lie groupoids more precisely we develop the theory of connections on lie groupoid g extensions which we call connections on gerbes and study the induced connections on various associated bundles we also prove analogues of the bianchi identities in particular we develop a cohomology theory which measures the existence of connections and curvings for g gerbes over stacks we also introduce g central extensions of groupoids generalizing the standard groupoid s 1 central extensions as an example we apply our theory to study the differential geometry of g gerbes over a manifold

multiple comparisons procedures for comparing several treatments with a control based on binary data in this paper we examine three approaches for comparing several treatments with a control with use of binary response data the first approach relies on asymptotic theory applied to the freeman tukey transformation of the observed proportions the second finds an acceptance region based on the binomial distributions estimated under the joint null hypotheses the third approach applies dunnett s procedure to the binary data we evaluated the actual overall type i error rates of the freeman tukey test and dunnett s procedure using both simulation and binomial calculations while we assessed those of the binomial approach using simulation based upon their capability to preserve the desirable overall type i error rate we provide recommendations regarding the choices among the three approaches for various occasions in addition we provide comments on the power of these three approaches

on the relative dual of an s1 gerbe over an orbifold abstract we construct a new effective orbifold y with an s 1 gerbe c to study an s 1 gerbe t on a g gerbe y over an orbifold b we view the former as the relative dual relative to b of the latter we show that the two pairs y t and y c have isomorphic categories of sheaves and also the associated twisted groupoid algebras are morita equivalent as a corollary the k theory and cohomology groups of y t and y c are isomorphic

asymptotic controllability and observability imply semiglobal practical asymptotic stabilizability by sampled data output feedback it is shown for general nonlinear systems that asymptotic controllability and observability are sufficient for semiglobal practical asymptotic stabilization by output feedback indeed as previously shown in the literature asymptotic controllability implies the existence of a discontinuous in general state feedback that when implemented by sample and hold is semiglobally practically stabilizing and robust to measurement disturbance moreover a weak form of observability allows reconstruction of the state with arbitrary precision in an arbitrarily short amount of time so we can build an output feedback that operates periodically in two modes an initial small fraction of a sampling period is used to estimate the state and the remainder of the sampling period is used to implement the state feedback control using the state estimate our stabilization results are presented not only for compact target sets e g the origin but also for noncompact target sets

differential geometry of gerbes abstract we define in a global manner the notion of a connective structure for a gerbe on a space x when the gerbe is endowed with trivializing data with respect to an open cover of x we describe this connective structure in two separate ways which extend from abelian to general gerbes the corresponding descriptions due to j l brylinski and n hitchin we give a global definition of the 3 curvature of this connective structure as a 3 form on x with values in the lie stack of the gauge stack of the gerbe we also study this notion locally in terms of more traditional lie algebra valued 3 forms the bianchi identity which the curvature of a connection on a principal bundle satisfies is replaced here by a more elaborate equation