super lie groups global topology and local structure a general mathematical framework for the super lie groups of supersymmetric theories is presented the definition of super lie group is given in terms of supermanifolds and two theorems analogous to theorems in classical lie group theory are proved the relationship of the super lie groups defined here to the formal groups of berezin and kac and the graded lie groups of kostant is analyzed

even sectors of lie superalgebras as locally convex lie algebras let g be a finite dimensional lie superalgebra over a graded commutative local augmented arens michael ground algebra a lie group is associated with a number of pleasant properties to the even sector g0 viewed as an ordinary complete locally convex lie algebra which is in general infinite dimensional

superspaces and supersymmetries a theory of graded banach modules over a banach grassmann algebra is developed and applied to differential geometry of super manifolds the explicit structure of superspaces carrying poincare supersymmetry and extended supersymmetry including central charges is described

confidence and conflict in multivariate calibration summary multivariate calibration involves using an estimated relationship between a multivariate response y and an explanatory vector x to predict unknown x in future from further observed responses in controlled calibration with multivariate normal errors the profile likelihood function for the unknown x denoted 4 displays a term which measures the mutual inconsistency of the given response vector denoted z in predicting 4 this inconsistency diagnostic fundamentally differentiates the behaviour of likelihood based and bayes confidence regions from those of the unconditional sampling approach in addition the diagnostic serves to pinpoint an inadequate response vector z

contractions of lie algebras with 2 dimensional generic coadjoint orbits abstract we determine all the contractions within the class of finite dimensional real lie algebras whose coadjoint orbits have dimensions 2

a comparison of uncertainty criteria for calibration calibration consists of using a fitted regression line to estimate the value of an unobserved independent variable x corresponding to an observed dependent variable y to construct a confidence interval for a single x eisenhart introduced a procedure that consists of inverting prediction intervals around the regression line numerous other inference procedures have been proposed for multiple use calibration in which a single fitted regression line is used repeatedly to estimate many x s we provide a synthesis of this literature and offer some numerical comparisons we also attempt to motivate the use of various criteria based on the particular points of view of the various parties involved in determining the calibration or using the results in addition we derive probability expressions for computing exact simultaneous prediction intervals that enable the construction of tighter limits than are currently available based on that criterion

a quick and easy multiple use calibration curve procedure the standard multiple use calibration procedure of scheffe 1973 states that with probability 1 the proportion of calculated confidence intervals containing the true unknowns is at least 1 in the long run the probability 1 refers to the probability that the calibration experiment results in a good outcome in scheffe s formulation a good outcome involves both coverage of the true underlying regression curve and an upper confidence limit for the scale parameter scheffe s procedure is fairly difftcult for practitioners to apply because it relies on tables that are not easy to use a simpler notion of goodness that requires only the coverage of the underlying regression leads to easily calculated confidence intervals for the unknowns in addition these intervals are generally shorter than scheffe s an application example is given to illustrate the technique

estimating fetal age by ultrasound measurements an example of multivariate calibration a problem in the estimation of gestational age by ultrasound measurement of fetal bone lengths the femur length f and the biparietal diameter bpd is described by means of multivariate calibration techniques f and bpd are combined to give an index of gestational age which is significantly more accurate than existing indices based on f or bpd alone a detailed description of the statistical analysis is given ranging from the initial steps of data analysis and model choice to the final step of determining the index and its accuracy a number of interesting aspects of multivariate calibration theory are illustrated

global properties of supermanifolds we construct new examples of supermanifolds and determine the vector bundle structure of the supermanifolds commonly used in physics we show that any supermanifold admits a foliation whose leaves are locally tangent to the soul directions in the coordinate charts and which is one of a nested sequence of foliations we point out that the existence of these foliations implies restrictions on the possible topologies of supermanifolds for example a compact supermanifold with a single even dimension must have vanishing euler characteristic we also show that a globally defined superfield on a nice compact supermanifold must be constant along the leaves of the foliations by this mechanism the global topology of a supermanifold can be used to impose physically interesting constraints on superfields as an example we exhibit a supermanifold which has the local geometry of flat superspace but is such that all globally defined superfields are chiral

super riemann surfaces uniformization and teichmuller theory teichmuller theory for super riemann surfaces is rigorously developed using the supermanifold theory of rogers in the case of trivial topology in the soul directions relevant for superstring applications the following results are proven the super teichmuller space is a complex super orbifold whose body is the ordinary teichmuller space of the associated riemann surfaces with spin structure for genusg 1 it has 3g 3 complex even and 2g 2 complex odd dimensions the super modular group which reduces super teichmuller space to super moduli space is the ordinary modular group there are no new discrete modular transformations in the odd directions the boundary of super teichmuller space contains not only super riemann surfaces with pinched bodies but rogers supermanifolds having nontrivial topology in the odd dimensions as well we also prove the uniformization theorem for super riemann surfaces and discuss their representation by discrete supergroups of fuchsian and schottky type and by beltrami differentials finally we present partial results for the more difficult problem of classifying super riemann surfaces of arbitrary topology

the theory of g supermanifolds on developpe une theorie des supervarietes dans laquelle une supervariete est une variete ordinaire associee a une g structure d ordre 2 integrable on demontre un theoreme de structure

supermanifold cohomology and the wess zumino term of the covariant superstring action the cohomology theory of supermanifolds is developed its basic properties are established and simple examples given the wess zumino term in the green schwarz covariant superstring action is interpreted as a nontrivial class in the supersymmetric cohomology of flat superspace a quotient supermanifold with nontrivial topology reflecting this class is constructed it is shown that there is no topological quantization condition for the coefficient of the wess zumino term the superstring differs from conventional sigma models in this respect because its action is grassmann valued and its group manifold superspace is noncompact

discrete supergroups and super riemann surfaces the properties of discrete supergroups that represent the homotopy groups of super riemann surfaces are emphasized after the study of the superforms invariant with respect to these supergroups the set of superconformal differentials are defined and an example is given of those by defining poincare s super theta functions some remarks on a possible definition of the jacobi variety of a super riemann surface are given

generalized supermanifolds i superspaces and linear operators generalized superspaces are defined taking as a starting point the concept of grassmann banach algebra the category of b spaces with lb maps as morphisms is introduced this allows one to discuss linear and tensor algebra on generalized superspaces

on lp multipliers of cartan motion groups abstract let g k be a riemannian symmetric pair of the compact type and let v k be the associated cartan motion group we establish a body of approximation theorems that permit the transfer of fourier analysis from the group g to the cartan motion group as a consequence we prove an analogue of a theorem of de leeuw concerning restriction of fourier multipliers of the group r to the group z the latter theorem extends an earlier result of r l rubin for the case where g so 3 k so 2 and v k is the euclidean motion group

enlargeable graded lie algebras of supersymmetry a criterion to enlarge infinite dimensional lie algebras to analytic lie groups is used here for the extension of graded lie algebras of physical interest to super lie groups

confidence regions in multivariate calibration on propose une methode alternative qui donne une region a bon comportement correspondant a la region invariante par translation uniformement la plus precise

multiple use confidence regions in multivariate calibration abstract the problem of multivariate calibration is considered in the setup where a normally distributed response variable is related to an explanatory variable through a multivariate linear model the variance covariance matrix of the response variable is assumed to be a multiple of the identity matrix the calibration data that is data obtained on the response variable corresponding to known values of the explanatory variable are to be used for the construction of confidence regions for unknown values of the explanatory variable the calibration problem addressed in this article deals with the construction of multiple use confidence regions that is the calibration data will be used repeatedly in order to construct a sequence of confidence regions for a sequence of unknown values of the explanatory variable such a procedure is characterized using two coverage probabilities say 1 and 1 given that the confidence regions are constructed using the same calibration data the proportion of conf

bootstrapping in controlled calibration experiments we consider the determination of an unknown quantity for example the concentration of a particular chemical in a given sample or samples using controlled calibration here several samples are prepared with concentrations chosen to cover a required range and these are used to establish the relationship between concentration and the measured response to an assay method this relationship is then used to estimate the concentration in the unknown samples from their measured responses confidence intervals for the estimated concentrations can usually be calculated by inverting a prediction interval but in some situations this method becomes intractable we explore the use of the bootstrap as an alternative in linear nonlinear and multivariate controlled calibration using both simulation and real datasets from the field of immunoassay we also discuss the alternatives afforded by replication of the design points the bootstrap is found to be comparable to the standard method in simple situations and is ea

pointwise and simultaneous tolerance limits under logistic regression this article investigates computation of pointwise and simultaneous tolerance limits under the logistic regression model for binary data the data consist of n binary responses where the probability of a positive response depends on covariates via the logistic regression function upper tolerance limits are constructed for the number of positive responses in m future trials for fixed as well as varying levels of the covariates the former provides pointwise upper tolerance limits and the latter provides simultaneous upper tolerance limits the upper tolerance limits are obtained from upper confidence limits for the probability of a positive response modeled using the logistic function to compute pointwise upper confidence limits for the logistic function likelihood based asymptotic methods small sample asymptotics as well as bootstrap methods are investigated and numerically compared to compute simultaneous upper tolerance limits a bootstrap approach is investigated the problems have been motiva

simultaneous tolerance intervals for the linear regression model abstract statistical tolerance intervals are developed for the normal regression model these intervals are constructed to guarantee at least p content for all possible values of the predictor variates the confidence set approach suggested by wilson 1967 is used wilson used an ellipsoidal confidence set for the regression coefficients and standard deviation of the residuals which imposes an unnecessary lower bound on by modifying the ellipsoidal confidence set to remove the lower bound imposed on we obtain narrower tolerance intervals another confidence set formed from the product set of confidence sets for and is used to construct tolerance intervals the tolerance intervals of the two new procedures are compared with those of the wilson method for a simple linear regression example the tolerance intervals based on the product confidence set are found to be efficient and easy to compute compared with those constructed from the ellipsoidal and the modified ellipsoidal confidence sets

symmetries of the hydrogen atom and algebraic families we show how the schrodinger equation for the hydrogen atom in two dimensions gives rise to an algebraic family of harish chandra pairs that codifies hidden symmetries the hidden symmetries vary continuously between so 3 so 2 1 and the euclidean group o 2 r2 we show that the solutions of the schrodinger equation may be organized into an algebraic family of harish chandra modules furthermore we use jantzen filtration techniques to algebraically recover the spectrum of the schrodinger operator this is a first application to physics of the algebraic families of harish chandra pairs and modules developed in the work of bernstein et al int math res notices rny147 2018 rny146 2018 we show how the schrodinger equation for the hydrogen atom in two dimensions gives rise to an algebraic family of harish chandra pairs that codifies hidden symmetries the hidden symmetries vary continuously between so 3 so 2 1 and the euclidean group o 2 r2 we show that the solutions of the schrodinger equation may be organized into an algebraic family of harish chandra modules furthermore we use jantzen filtration techniques to algebraically recover the spectrum of the schrodinger operator this is a first application to physics of the algebraic families of harish chandra pairs and modules developed in the work of bernstein et al int math res notices rny147 2018 rny146 2018

on n 4 super riemann surfaces and the superconformal semigroup the finite n 4superconformal scf transformations areobtained and classified into three types by means of the permanent of specific matrices the properties of the latter are studied in connection with complex plane non euclidean geometry the explicit expression for the n 4 scf transformations allows them to be used as the transition functions for n 4 split and non split super riemann surfaces and scf embeddings the existence of the semigroup of scf transformations is indicated

supertori are algebraic curves super riemann surfaces of genus 1 with arbitrary spin structures are shown to be the sets of zeroes of certain polynomial equations in projective superspace we conjecture that the same is true for arbitrary genus properties of superelliptic functions and super theta functions are discussed the boundary of the genus 1 super moduli space is determined

on the semigroup nature of superconformal symmetry a semigroup of n 1 superconformal transformations is introduced and analyzed noninvertible ones can describe transitions from body to soul and form a proper ideal containing a set of nilpotent transformations the projective superspace is also considered transformations twisting the parity of a tangent space are brought in they can be a nonsuperconformal square root of noninvertible superconformal transformations and the analogs of the poincare metric and conformal invariance are suggested for them

wavelets on the n sphere and related manifolds we present a purely group theoretical derivation of the continuous wavelet transform cwt on the n 1 sphere sn 1 based on the construction of general coherent states associated to square integrable group representations the parameter space of the cwt x similar to so n xr is embedded into the generalized lorentz group so0 n 1 via the iwasawa decomposition so that x similar or equal to so0 n 1 in where n similar or equal to rn 1 then the cwt on sn 1 is derived from a suitable unitary representation of so0 n 1 acting in the space l 2 sn 1 d mu of finite energy signals on sn 1 which turns out to be square integrable over x we find a necessary condition for the admissibility of a wavelet in the form of a zero mean condition which entails all the usual filtering properties of the cwt next the euclidean limit of this cwt on sn 1 is obtained by redoing the construction on a sphere of radius r and performing a group contraction for r infinity from which one recovers the usual cwt on flat euclidean space finally we discuss the extension of this construction to the two sheeted hyperboloid hn 1so0 n 1 1 so n 1 and some other riemannian symmetric spaces c 1998 american institute of physics s0022 2488 98 00308 9

errors in variables estimation in multivariate calibration a set of q responses y y 1 y 2 y q t is related to a set of p explanatory variables x x 1 x 2 x p t through the classical linear regression model y t a t x t b e t first the unknown parameters a and b are estimated using a calibrafion set the statistical problem that is considered here is that of estimating the vector x o that underlies a new observed vector of responses y o using the parameter estimates obtained from the first procedure these two procedures are commonly referred to as calibration and prediction or inverse prediction and sometimes jointly referred to as calibration the prediction procedure can be viewed as parameter estimation in errors in variables regression the maximum likelihood estimator assuming normally distributed measurement errors is proposed for the prediction procedure unlike the classical estimator used in the prediction procedure the proposed estimator is consistent with respect to the number of response variables the performan

how different are the supermanifolds of rogers and dewitt a dewitt supermanifold always has the structure of a vector bundle over an ordinary spacetime manifold whereas a rogers supermanifold is not so restricted corresponding to the vector space fibers of the dewitt supermanifold a rogers supermanifold has a foliation by submanifolds or leaves parametrized by soul coordinates only we show that the universal covering space of any leaf always admits a flat metric if the covering space is complete in this metric it must in fact be a vector space we combine this result with known theorems about foliations to give conditions under which a compact rogers supermanifold with a single even dimension is necessarily a quotient space of flat superspace we also show that a supermanifold defined by a polynomial equation in flat superspace is always of the dewitt type finally we exhibit new supermanifold structures forr2 and the 2 torus which show that the foliation of a rogers supermanifold can be quite exotic

on contractions of semisimple lie groups a limiting formula is given for the representation theory of the cartan motion group associated to a riemannian symmetric pair g k in terms of the representation theory of g introduction let g be a connected lie group with lie algebra g and h a closed subgroup with subalgebra b the coset space g h is called reductive 9 p 389 if h admits an adg h invariant complement m in g i e a subspace m c g such that i g b m direct sum ii adg h m c m in this case we can form the semidirect product m a h with respect to the adjoint action of h on m in this paper we shall restrict ourselves to the case where g is semisimple with finite centre and g h is a riemannian symmetric pair 10 p 209 hence h is contained in the fixed point set ha of an analytic involution a of g it contains the identity component ha e and adc h is compact following custom we write k and f rather than h and b in this instance it is well known that k is compact 10 p 252 and connected if g is noncompact furthermore f is the 1 eigenspace of doe we make the natural choice for m namely the 1 eigenspace v of dae when g is noncompact dae is a cartan involution then v is usually denoted p and g f is called a cartan decomposition 10 p 182 when g is compact one can choose a real form g0 of the complexification gc of g and a cartan

calibration and simultaneous tolerance intervals for regression simultaneous calibration or discrimination intervals in regression were proposed by lieberman miller and hamilton 1967 and by scheffe 1973 those procedures enable one to construct confidence intervals for the unobserved values of the independent variable corresponding to an unlimited sequence of observations of the dependent variable in a regression model these calibration intervals are conservative in that they are obtained from simultaneous tolerance intervals for which the actual confidence level exceeds the nominal level furthermore all other existing simultaneous tolerance intervals in regression are likewise conservative in this article we propose simultaneous tolerance intervals that are narrower than previous intervals given the tables of factors included in this article they are also simple to construct and use in straightline calibration applications

conservative confidence regions in multivariate calibration in the multivariate calibration problem using a multivariate linear model some conservative confidence regions are constructed the regions are nonempty and invariant under nonsingular transformations situations where the explanatory variable occurs nonlinearly in the model are also considered computational aspects of the confidence region and its practical implementation are discussed the results are illustrated using two examples the examples show that our confidence regions are much more satisfactory compared to those based on some of the existing procedures furthermore simulation results for the examples reveal that the coverage probability of the conservative confidence regions are very close to the assumed confidence level

berezin integration on general fermionic supermanifolds recent results on the global structure of supermanifolds are used to define a notion of berezin integration on any purely fermionic rogers supermanifold this leads to an integration theory on a large class of supermanifolds having both bosonic and fermionic coordinates the existence of global functions and forms on such supermanifolds is discussed as is some elementary cohomology of supermanifolds

gabor analysis as contraction of wavelets analysis we use the method of group contractions to relate wavelet analysis and gabor analysis wavelet analysis is associated with unitary irreducible representations of the affine group while the gabor analysis is associated with unitary irreducible representations of the heisenberg group we obtain unitary irreducible representations of the heisenberg group as contractions of representations of the extended affine group furthermore we use these contractions to relate the two analyses namely we contract coherent states resolutions of the identity and tight frames in order to obtain the standard gabor frame we construct a family of time localized wavelet frames that contract to that gabor frame starting from a standard wavelet frame we construct a family of frequency localized wavelet frames that contract to a nonstandard gabor frame in particular we deform gabor frames to wavelet frames

soul expansion of g superfunctions supersmooth or g superfunctions with coefficients in the banach grassmann rogers algebra b are described in terms of the soul expansion and the notion of pringsheim regular real function

a nonparametric calibration analysis in this paper we discuss a new approach to solve calibration problems in a nonparametric setting this approach is appealing because it yields estimates of the required quantities directly the method combines kernel and robust estimation techniques it relies on strong approximations of the estimating process and the extreme value theorem of bickel and rosenblatt using these results we first obtain robust pointwise estimates of the parameters of interest second we set up asymptotic simultaneous tolerance regions for many unknown values of the quantity to be calibrated the technique is illustrated on a radiocarbon dating problem the nonparametric calibration procedure proves to be of practical as well as theoretical interest moreover it is quick and simple to implement 1 introduction statistical calibration analyses provide a way to predict a quantity from the observation of another one by using a dose response type relationship the problem occurs in industry or the biological sciences when the quantity to be calibrated is hard or expensive to measure or is not observable in many situations the knowledge of the experimental process can hardly be translated into a parametric model for the dose response relationship a tempting alternative may be provided by a nonparametric modeling let z denote the random response variable we observe the unknown

graded manifolds supermanifolds and infinite dimensional grassmann algebras a new approach to superdifferentiable functions of grassmann variables is developed which avoids ambiguities in odd derivatives this is used to give an improved definition of supermanifold over a finite dimensional grassmann algebra a natural embedding of super manifolds over grassmann algebras with increasing number l of generators is developed and thus a limit asl tends to infinity is possible a correspondence between graded manifolds and supermanifolds is constructed extending results of 5 and 8

generalized supermanifolds ii analysis on superspaces the analysis on generalized superspaces is studied g differentiability of functions and indetermination of partial g derivatives are treated in detail a canonical expansion for supersmooth functions and a particular choice of partial g derivatives are given and their properties studied existence of nonanalytic g functions is proved

generalized supermanifolds iii ρ supermanifolds supermanifolds are characterized as supermanifolds modeled in a superspace s having a well behaved body tangent cotangent and tensor bundles to supermanifolds are discussed the elements of the theory of lie supergroups and principal superfiber bundles as supermanifolds are given

noninvertible n 1 superanalog of complex structure we consider an alternative tangent space reduction in n 1 superspace which leads to some odd n 1 superanalog of complex structure the even one is widely used in two dimensional superconformal theories and in the fermionic string theory calculations via super riemann surfaces new n 1 superconformal like transformations are similar to the antiholomorphic ones of the complex function theory they are dual to the ordinary superconformal transformations subject to the berezinian addition formula presented noninvertible highly degenerated and twist parity of the tangent space in the standard basis and they also lead to some mixed cocycle condition a new parametrization for the superconformal group is presented which allows us to extend it to a semigroup and to unify the description of old and new transformations

on the analogy between real reductive groups and cartan motion groups contraction of irreducible tempered representations attached to any reductive lie group g is a cartan motion group g 0 a lie group with the same dimension as g but a simpler group structure a natural one to one correspondence between the irreducible tempered representations of g and the unitary irreducible representations of g 0 whose existence had been suggested by mackey in the 1970s has recently been described by the author in the present notes we use the existence of a family of groups interpolating between g and g 0 to realize the bijection as a deformation for every irreducible tempered representation pi of g we build in an appropriate fr echet space a family of subspaces and evolution operators that contract pi onto the corresponding representation of g 0

redundancy of the quantum level gauge condition the manifold defined by the equations of motion em of the gauge and ghost fields w r t the gauge fixed action is regarded as a supercanonical line bundle over the manifold defined by the em of the gauge fields only w r t the classical action in this language the brst operator q is a local section of the bundle using this we give a local expression for q as being on the other hand the nilpotent exterior derivative on with the ghost field as its generator this fiber bundle setup allows us to prove that any second level gauge condition i e a gauge condition on is equivalent to a gauge on the base manifold and thus does not break the brst symmetry of the quantized theory