the duality in octonions and the triality principle we show that the existence of an a 2 type subalgebra of e 6 which is dual to another one of g 2 type leads to a definition of the product of octonions on an 8 dimensional subspace of e 6 thisproduct is then only expressed in terms of the lie bracket of e 6 the well knowntriality principle becomes an easy consequence of this definition and g 2 acting by the adjoint action isshown to be the algebra of derivations of the octonions the real octonions are obtained from two specific real forms of e 6

several methods to assess improvement in risk prediction models extension to survival analysis risk prediction models have been widely applied for the prediction of long term incidence of disease several parameters have been identified and estimators developed to quantify the predictive ability of models and to compare new models with traditional models these estimators have not generally accounted for censoring in the survival data normally available for fitting the models this paper remedies that problem the primary parameters considered are net reclassification improvement nri and integrated discrimination improvement idi we have previously similarly considered a primary measure of concordance area under the roc curve auc also called the c statistic we also include here consideration of population attributable risk par and ratio of predicted risk in the top quintile of risk to that in the bottom quintile we evaluated estimators of these various parameters both with simulation studies and also as applied to a prospective study of coronary heart disease chd our simulation studies showed that in general our estimators had little bias and less bias and smaller variances than the traditional estimators we have applied our methods to assessing improvement in risk prediction for each traditional chd risk factor compared to a model without that factor these traditional risk factors are considered valuable yet when adding any of them to a risk prediction model that has omitted the one factor the improvement is generally small for any of the parameters this experience should prepare us to not expect large values of the risk prediction improvement evaluation parameters for any new risk factor to be discovered

toeplitz operators and solvable c algebras on hermitian symmetric spaces bounded symmetric domains cartan domains and exceptional domains are higher dimensional generalizations of the open unit disc in this note we give a structure theory for the c algebra t generated by all toeplitz operators tf h p fh with continuous symbol function g c s on the shilov boundary 5 of a bounded symmetric domain d of arbitrary rank r here h belongs to the hardy space h s and p l s h s is the szego projection for domains of rank 1 and tube domains of rank 2 the structure of t has been determined in 1 2 in these cases toeplitz operators are closely related to pseudodifferential operators for the open unit disc t is the c algebra generated by the unilateral shift the structure theory for the general case 12 is based on the fact that d can be realized as the open unit ball of a unique jordan triple system z 7 theorem 4 1 denoting the jordan triple product by uv w a tripotent e g z satisfies ee e e tripotents generalize the partial isometries of matrix algebras and determine the boundary structure of d c z cf 7 theorem 6 3 our principal result 12 cf also 3 4 8 is the following

toeplitz c algebras over pseudoconvex reinhardt domains this is the published version also available here http dx doi org 10 1090 s0273 0979 1989 15753 4 first published in bull amer math soc in 1989 published by the american mathematical society

spectrum and analytical indices of the c algebra of wiener hopf operators we study multivariate generalisations of the classical wiener hopf algebra which is the c algebra generated by the wiener hopf operators given by convolutions restricted to convex cones by the work of muhly and renault this c algebra is known to be isomorphic to the reduced c algebra of a certain restricted action groupoid given by the action of euclidean space on a certain compactification using groupoid methods we construct composition series for the wiener hopf c algebra by a detailed study of this compactification we compute the spectrum and express homomorphisms in k theory induced by the symbol maps which arise by the subquotients of the composition series in analytical terms namely these symbols maps turn out to be given by an analytical family index of a continuous family of fredholm operators in a subsequent paper we also obtain a topological expression of these indices

the capelli identity the double commutant theorem and multiplicity free actions 0 the capelli identity cal 3 w p 39 is one of the most celebrated and useful formulas of classical invariant theory w d cl z the double commutant theorem w p 91 is likewise a basic result in the general theory of associative algebras both play key roles in weyl s book the classical groups the main purpose of this paper is to demonstrate a close connection between the two in the context of multiplicity free actions ekj of groups on vector spaces the focus of our discussion will be the structure of the differential operators which commute with a multiplicity free action applications include a new derivation of some formulas of shimura shl and rubenthaler and schiffmann rs for b functions associated to hermitian symmetric spaces and a construction of interesting sets of generators for the center of the universal enveloping algebra of gin we also give a detailed discussion of certain aspects of multiplicity free representations here is an overview of the contents of the paper in sect 1 we review the classical capelli identities we observe how their existence is predicted by a double commutant result 1 6 and 1 8 and we show how they can be used to compute b functions we remark that capelli understood that his operators generate the center of the algebra generated by the polarization operators see b1 p 77 also capelli s motivation in introducing his operators was to explain a formula of cayley essentially the computation of a b function capelli s point of view was remarkably modern and structuralist in certain ways more modern even than that of weyl sections 2 through 9 provide a conceptual context for understanding the capelli identities as a feature of multiplicity free actions they contain a general discussion of the structure of g the algebra of polynomial coefficient differential operators which commute with a given group g of linear transformations the results of the discussion are summarized in theorem 9 1 which says in particular that the polynomial coefficient differential operators commuting with a

singular invariant tempered distributions on regular prehomogeneous vector spaces revue des espaces vectoriels prehomogenes distributions temperees invariantes distributions temperees avec un parametre meromorphe puissances complexes des invariants relatifs soit p x un polynome homogene irreductible a coefficients reels sur un espace vectoriel v soit s x v p x 0 se decompose en un nombre fini de g 1 orbites avec g 1 g gl v p g x p x on pose v 1 uv 2 u uv e v s la decomposition en composantes connexes on donne une condition necessaire et suffisante pour qu une distribution temperee g 1 invariante a support dans s puisse s obtenir comme une combinaison lineaire finie de coefficients de laurent negatifs de la distribution p x s vd i

survival model predictive accuracy and roc curves summary the predictive accuracy of a survival model can be summarized using extensions of the proportion of variation explained by the model or r2 commonly used for continuous response models or using extensions of sensitivity and specificity which are commonly used for binary response models in this article we propose new time dependent accuracy summaries based on time specific versions of sensitivity and specificity calculated over risk sets we connect the accuracy summaries to a previously proposed global concordance measure which is a variant of kendall s tau in addition we show how standard cox regression output can be used to obtain estimates of time dependent sensitivity and specificity and time dependent receiver operating characteristic roc curves semiparametric estimation methods appropriate for both proportional and nonproportional hazards data are introduced evaluated in simulations and illustrated using two familiar survival data sets

c algebras of multiplication operators on bergman spaces soit un domaine borne dans c soit a 2 l espace de bergman de toutes les fonctions analytiques sur qui sont de carre integrable par rapport a la mesure volumetrique et soit c la c algebre engendree par les operateurs de multiplication par les fonctions coordonnee sur on etudie les proprietes de c

novel metrics for evaluating improvement in discrimination net reclassification and integrated discrimination improvement for normal variables and nested models net reclassification and integrated discrimination improvements have been proposed as alternatives to the increase in the auc for evaluating improvement in the performance of risk assessment algorithms introduced by the addition of new phenotypic or genetic markers in this paper we demonstrate that in the setting of linear discriminant analysis under the assumptions of multivariate normality all three measures can be presented as functions of the squared mahalanobis distance this relationship affords an interpretation of the magnitude of these measures in the familiar language of effect size for uncorrelated variables furthermore it allows us to conclude that net reclassification improvement can be viewed as a universal measure of effect size our theoretical developments are illustrated with an example based on the framingham heart study risk assessment model for high risk men in primary prevention of cardiovascular disease

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

homogeneous vector bundles and intertwining operators for symmetric domains the main features of homogeneous cowen douglas operators well known for the unit disk are generalised to the setting of hermitian bounded symmetric domains of arbitrary rank c 2016 elsevier inc all rights reserved

racines orthogonales et orbites d algebres de lie semi simple graduees abstract we define a particular class of regular prehomogeneous vector spaces of parabolic type in relation with orthogonal roots on a field of characteristic 0 we give the structure and the orbits of simple elements associated to the nonzero nilpotent elements of the prehomogeneous vector spaces in terms of these orthogonal roots

super toeplitz operators and non perturbative deformation quantization of supermanifolds the purpose of this paper is to construct non perturbative deformation quantizations of the algebras of smooth functions on poisson supermanifolds for the examplesu1 1 andcm n algebras of super toeplitz operators are defined with respect to certain hilbert spaces of superholomorphic functions generators and relations for these algebras are given the algebras can be thought of as algebras of quantized functions and deformation conditions are proven which demonstrate the recovery of the super poisson structures in a semi classical limit

a classification of 2 simple prehomogeneous vector spaces of type i let p g gl v be a rational representation of a connected linear algebraic group g on a finite dimensional vector space v all defined over an algebraically closed field k of characteristic zero if v has a zariskidense g orbit we call a triplet g p v a prehomogeneous oector space abbrev p v when p is irreducible such p v s have been classified in 11 since then it has turned out gradually that the complete classification of reductive p v s i e p v s with reductive groups g is an extremely laborious task therefore it is natural to classify some restricted class of p v s e g 2 to get some insight into the general situation a p v g p v is called a 2 simple p v when 1 g gl 1 x g x g with simple algebraic groups g and gz 2 p is the composition of a rational representation p of g x g of the form p p1 0 p pkol a a l l z t with k s t l where pi tv resp pi t are nontrivial irreducible representations of g resp g and the scalar multiplications gz 1 on each irreducible component v for i 1 f where v i v we say that a 2 simple p v g p v is of type zif k l and at least one of gl l xg xg pi pi i 1 k is a nontrivial p v see definition 5 p 43 in 11 on the other hand if k l and all gl l xg xg p op i l k are trivial p v s it is called a 2 simple p v of type ii in 3 all 2 simple p v s of type ii has been already classified in this paper we shall classify all 2simple p v s of type i thus together with 3 we complete a classification of all 2 simple p v s for example the fact that all irreducible p v s are castling equivalent to 2 simple p v s or to x m x y m x g 2 1 1 i i n 1 with m 2 3 see 1 indicates the importance of 2 simple p v s for simplicity we write g p v or g p instead of g p v

racines orthogonales formes quadratiques et orbites d algebres de lie simples graduees abstract we give a unified orbital decomposition for irreducible regular prehomogeneous vector spaces of parabolic type graded by strongly orthogonal roots of same length with one dimensional corresponding root space on a local or global field of characteristic 0 in terms of a class of quadratic forms related to a root system weyl action and the invariants of the prehomogeneous vector spaces

the partial area under the summary roc curve the area under the curve auc is commonly used as a summary measure of the receiver operating characteristic roc curve it indicates the overall performance of a diagnostic test in terms of its accuracy at various diagnostic thresholds used to discriminate cases and non cases of disease the auc measure is also used in meta analyses where each component study provides an estimate of the test sensitivity and specificity these estimates are then combined to calculate a summary roc sroc curve which describes the relationship between test sensitivity and specificity across studies the partial auc has been proposed as an alternative measure to the full auc when using the partial auc one considers only those regions of the roc space where data have been observed or which correspond to clinically relevant values of test sensitivity or specificity in this paper we extend the idea of using the partial auc to sroc curves in meta analysis theoretical and numerical results describe the variation in the partial auc and its standard error as a function of the degree of inter study heterogeneity and of the extent of truncation applied to the roc space a scaled partial area measure is also proposed to restore the property that the summary measure should range from 0 to 1 the results suggest several disadvantages of the partial auc measures in contrast to earlier findings with the full auc the partial auc is rather sensitive to heterogeneity comparisons between tests are more difficult especially if an empirical truncation process is used finally the partial area lacks a useful symmetry property enjoyed by the full auc although the partial auc may sometimes have clinical appeal on balance the use of the full auc is preferred

matrix cartan superdomains super toeplitz operators and quantization we present a general theory of non perturbative quantization of a class of hermitian symmetric supermanifolds the quantization scheme is based on the notion of a super toeplitz operator on a suitable z2 graded hilbert spaces of super holomorphic functions the quantized supermanifold arises as the c algebra generated by all such operators we prove that our quantization framework reproduces the invariant super poisson structure on the classical supermanifold as planck s constant tends to zero

arithmetic calculus of fourier transforms by igusa local zeta functions we show the possibility of explicit calculation of the fourier transforms of complex powers of relative invariants of some prehomogeneous vector spaces over jr by using the explicit form of p adic igusa local zeta functions let g p v be a regular prehomogeneous vector space defined over r and fi f the basic r relative invariants of g p v cf f sato 2 p 444 for explicit calculation of fourier transforms of the complex power r if x is 1 if1 x isi i 1 with s si sr e cr two general methods are known when g p v has finitely many g orbits one can use a microlocal calculus established by m kashiwara according to m sato s idea see p 404 in m k k m when r 1 and the relative invariant is linear for each variable one can use igusa s method see p 8 in igusa 2 in this paper we shall show the possibility of calculation for some cases by using the explicit form of the igusa s local zeta function based on the idea of iwasawa tate theory iwasawa tate as an example we shall calculate the fourier transform of if x is for gl4 x sl2m l a2 d al d a d a alt2m 1 d aff2m l d aff2m i 1 d aff2m which has infinitely many orbits and r 4 so that it has not been calculated by other methods the igusa local zeta function of this prehomogeneous vector space has been explicitly calculated by hosokawa our method is applicable for a prehomogeneous vector space with za rzm for some constant r 0 for example irreducible resp simple 2 simple of type i regular prehomogeneous vector spaces with finitely many adelic open orbits satisfy za rzm see igusa 4 and k k this paper has four sections 1 preliminaries 2 functional equation 3 unramified f factors and igusa local zeta functions 4 arithmetic calculus example the last section of this paper was completed when the author was received by the editors july 27 1993 and in revised form march 30 1994 1991 mathematics subject classification primary 32m1 2 20g20 secondary 1 1s40

the adelic zeta function associated with the space of binary cubic forms with coefficients in a function field in this paper we study the adelic zeta function associated with the prehomogeneous vector space of binary cubic forms defined over a function field we establish its rationality find its poles and residues and a simple functional equation that this zeta function satisfies introduction the subject of integral equivalence of binary cubic forms although not as popular or well developed as the theory of integral equivalence of binary quadratic forms has received a fair amount of attention the first paper on this topic 9 written by eisenstein appeared in 1844 later on several other first rate mathematicians tried their hand at the subject among others the list includes arndt cayley hermite and davenport the classical method of attacking this problem is to draw upon a one to one discriminant preserving relation between binary cubic and binary quadratic forms which assigns to a cubic form its hessian and then use the reduction theory for binary quadratic forms to obtain results on class numbers of binary cubic forms the best of the classical results are due to davenport 5 who showed that e h ir n 2 36x o x15 16 nax e h ir n v2 12x o x 5116 nax where h ir n is the class number of irreducible binary cubic forms of discriminant n the credit for a new surge of interest in the study of integral equivalence of binary cubic forms must indisputably go to shintani for it was his 1972 paper 25 that put this study on modern footing in that paper shintani introduced a certain zeta function which as it turns out decomposes into a linear combination of products of dirichlet series whose coefficients are weighted class numbers of binary cubic forms and certain mellin transforms by studying analytic properties of the zeta function shintani was able to improve the error term in davenport s result to o x576 it is of interest to note that the zeta function and the dirichlet series it received by the editors september 3 1985 1980 mathematics subject classification 1985 revision primary 12a70 secondary iocio 12a90 1987 american mathematical society 0002 9947 87 1 00 25 per page

quotients of some prehomogeneous vector spaces abstract we give quotients v g spec c v g for some prehomogeneous vector spaces g h v

measures for evaluation of prognostic improvement under multivariate normality for nested and nonnested models when comparing performances of two risk prediction models several metrics exist to quantify prognostic improvement including the change in the area under the receiver operating characteristic curve the integrated discrimination improvement the net reclassification index at event rate the change in standardized net benefit the change in brier score and the change in scaled brier score we explore the behavior and interrelationships between these metrics under multivariate normality in nested and nonnested model comparisons we demonstrate that within the framework of linear discriminant analysis all six statistics are functions of squared mahalanobis distance a robust metric that properly measures discrimination by quantifying the separation between the risk scores of events and nonevents these relationships are important for overall interpretability and clinical usefulness through simulation we demonstrate that the performance of the theoretical estimators under normality is comparable or superior to empirical estimation methods typically used by investigators in particular the theoretical estimators for the net reclassification index and the change in standardized net benefit exhibit less variability in their estimates as compared to their empirically estimated counterparts finally we explore how these metrics behave with potentially nonnormal data by applying these methods in a practical example based on the sex specific cardiovascular disease risk models from the framingham heart study our findings aim to give greater insight into the behavior of these measures and the connections existing among them and to provide additional estimation methods with less variability for the net reclassification index and the change in standardized net benefit

reinhardt domains boundary geometry and toeplitz c algebras abstract in this paper we have found explicitly how the boundary geometry of a reinhardt domain in c 2 determines the structure of its toeplitz c algebra more precisely our main result is an explicit simple algorithm to describe the structure of the toeplitz c algebra of a reinhardt domain d in c 2 satisfying some mild boundary condition in terms of rotation c algebras based on the degree of contact of the boundary curve of c the logarithmic domain of d at each point of intersection with the linear faces of the convex hull of c and the slopes of these faces a consequence of this result is that the toeplitz c algebras t d and t d of d and its pseudoconvex hull d are rarely isomorphic but are always of the same type i e either both are of type i or both are not of type i in other words the isomorphism class of t d is usually changed under taking pseudoconvex hull but the type of t d is not affected

minimal graded lie algebras and representations of quadratic algebras abstract let g 0 b 0 be a quadratic lie algebra i e a lie algebra g 0 with a non degenerate symmetric invariant bilinear form b 0 and let v be a finite dimensional representation of g 0 we define on g 0 b 0 v v g 0 v a structure of local lie algebra in the sense of kac 4 where the bracket between g 0 and v resp v is given by the representation resp and where the bracket between v and v depends on b 0 and this implies the existence of two z graded lie algebras g m a x g 0 b 0 v and g m i n g 0 b 0 v whose local part is g 0 b 0 v we investigate these graded lie algebras more specifically in the case where g 0 is reductive roughly speaking the map g 0 b 0 v g m i n g 0 b 0 v is a bijection between triplets and a class of graded lie algebras we show that the existence of associated sl 2 triples is equivalent to the existence of non trivial relative invariants on some orbit and we define the graded lie algebras of polynomial type which give rise to some dual airs

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

toeplitz algebras and rotational automorphisms associated to polydiscs introduction given a bounded symmetric domain d in cm with d being normalized to be circular and to contain the origin upmeier 22 showed that there is a natural way to associate a c algebra to it this c algebra called the toeplitz c algebra of d and denoted 35 is a solvable c algebra with a composition series jk o k n l of length n where n is the rank of the symmetric domain 22 theorem 3 12 we observe that there is a natural group action on 3 by the compact group g of all biholomorphic automorphisms of d which lie in the connected component of the identity and which fix the origin proposition 1 1 every element 0 in g preserves the solvable series jk o k n l of 3 d we thereby obtain a composition series jk x0 z for the crossed product c algebra 3 xo z in this paper we show by introducing the notion of stratification of composition series see section 2 that the series jk x0 z is exactly the maximal radical series of 3 x0 z section 3 and thus obtain an isomorphism invariant for the latter c algebra when is the polydisc d we determine the k theory of the series jk xo z and their subquotients section 4 techniques from k theory using strong morita equivalence yield a complete classification of the c algebras 3 xo z where d is the n dimensional polydisc up to isomorphism for all 0 in g tn

toeplitz c algebras on bounded symmetric domains in this paper we develop a structure theory for the toeplitz c algebra y for an arbitrary bounded symmetric domain d in c classical or exceptional as the main result it is shown that 5y is a c algebra of type i which is solvable i e admits a finite composition sequence of closed ideals sk in 7 such that the subquotients fk i k are continuous fields of elementary c algebras of compact operators the length of this composition sequence coincides with the rank r of d as a hermitian symmetric space for 0 k r the spectral components sk of ay parametrizing the irreducible representations of y are realized in terms of the boundary components complex analytic faces of d the structure theory for y is based on the fact that the tangent space z c c of d carries a jordan algebraic structure z is a jordan triple system with triple product uv w describing the geometry of d in algebraic terms for example d is the open unit ball with respect to the so called spectral norm on z and the boundary components of d are in 1 1 correspondence with the tripotents e e z partial isometries satisfying ee e 4 e in particular the shilov boundary s of d consists of all tripotents of maximal rank r the jordan theoretic description of d was used in 17 to determine an explicit peter weyl decomposition of the hardy space h2 s cf 15 in terms of the norm functions generalized determinants associated with the jordan triple

complete monotonicity for inverse powers of some combinatorially defined polynomials we prove the complete monotonicity on 0 infty n for suitable inverse powers of the spanning tree polynomials of graphs and more generally of the basis generating polynomials of certain classes of matroids this generalizes a result of szeg and answers among other things a long standing question of lewy and askey concerning the positivity of taylor coefficients for certain rational functions our proofs are based on two ab initio methods for proving that p beta is completely monotone on a convex cone c the determinantal method and the quadratic form method these methods are closely connected with harmonic analysis on euclidean jordan algebras or equivalently on symmetric cones we furthermore have a variety of constructions that given such polynomials can create other ones with the same property among these are algebraic analogues of the matroid operations of deletion contraction direct sum parallel connection series connection and 2 sum the complete monotonicity of p beta for some beta 0 can be viewed as a strong quantitative version of the half plane property hurwitz stability for p and is also related to the rayleigh property for matroids

dixmier trace for toeplitz operators on symmetric domains abstract for toeplitz operators on bounded symmetric domains of higher rank there is no obvious way to define the dixmier trace within the toeplitz c algebra since commutators are in general not compact in this paper we solve this problem by constructing a hilbert quotient module corresponding to partitions of length 1 which leads to commutators in the macaev class l n we also obtain an explicit formula for the dixmier trace of such commutators in terms of the underlying boundary geometry which involves a new type of flag manifold defined in jordan theoretic terms

quotients of standard hilbert modules we initiate a study of hilbert modules over the polynomial algebra that are obtained by completing with respect to an inner product having certain natural properties a standard hilbert module is a finite multiplicity version of one of these standard hilbert modules occupy a position analogous to that of free modules of finite rank in commutative algebra and their quotients by submodules give rise to universal solutions of nonlinear relations essentially all of the basic hilbert modules that have received attention over the years are standard including the hilbert module of the shift the hardy and bergman modules of the unit ball modules associated with more general domains in and those associated with projective algebraic varieties we address the general problem of determining when a quotient of an essentially normal standard hilbert module is essentially normal this problem has been resistant our main result is that it can be linearized in that the nonlinear relations defining the submodule can be reduced appropriately to linear relations through an iteration procedure and we give a concrete description of linearized quotients

discrimination slope and integrated discrimination improvement properties relationships and impact of calibration discrimination slope defined as the slope of a linear regression of predicted probabilities of event derived from a prognostic model on the binary event status has recently gained popularity as a measure of model performance it is as a building block for the integrated discrimination improvement that equals the difference in discrimination slopes between the two models being compared several authors have pointed out that it does not make sense to apply the integrated discrimination improvement and discrimination slope when working with mis calibrated models whereas others have raised concerns about the ability of improving discrimination slope without adding new information in this paper we show that under certain assumptions the discrimination slope is asymptotically related to two other r squared measures one of which is a rescaled version of the brier score known to be proper furthermore we illustrate how a simple recalibration makes the slope equal to the rescaled brier r squared metric we also show that the discrimination slope can be interpreted as a measure of reduction in expected regret for the gini brier regret function using theoretical and practical examples we illustrate how all of these metrics are affected by different levels of model mis calibration in particular we demonstrate that simple recalibration ascertaining calibration in the large and calibration slope equal to 1 are not sufficient to correct for some forms of mis calibration we conclude that r squared metrics including the discrimination slope offer an attractive choice for quantifying model performance as long as one accounts for their sensitivity to model calibration copyright 2016 john wiley sons ltd

the berezin transform and invariant differential operators the berezin calculus is important to quantum mechanics creation annihilation operators and operator theory toeplitz operators we study the basic berezin transform linking the contravariant and covariant symbol for all bounded symmetric domains and express it in terms of invariant differential operators

some p v equivalences and a classification of 2 simple prehomogeneous vector spaces of type a classification of 2 simple prehomogeneous vector spaces is completed by using some p v equivalences together with 3 some part is very different from the previous classification of the irreducible or simple cases 1 2 and some new method is necessary this result shows the difficult point of a classification problem of reductive prehomogeneous vector spaces introduction in m sato and t kimura 1 all irreducible prehomogeneous vector spaces abbreviated irreducible p v s are classified up to castling equivalence and it is proved that any irreducible p v is castling equivalent to a 2 simple p v or to sl m x sl m x gl 2 a1 x a1 x a1 with m 2 3 therefore as a step to a classification of the general case it is natural to classify all nonirreducible 2 simple p v s in this paper we shall classify all 2 simple p v s of type ii and give the complete list of them up to strong equivalence in the sense of definition 4 p 36 in 1 however we need various p v equivalences cf propositions 1 7 1 12 1 34 1 36 1 37 theorem 1 16 and its corollary corollary 1 26 definition 1 31 remark 1 32 to carry out the classification and hence first we shall prove them we also investigate the regularity in some cases since all 2 simple p v s of type i are already classified in 3 this completes the classification of all 2 simple p v s here we say that g p v or simply g p is a 2 simple p v if it is a prehomogeneous vector space of the following form g gl 1 k s t xg x g2 where g1 and g2 are simple algebraic groups p the composition of the scalar multiplications gl l k s on each irreducible component and the representation p p1 p 1 pk p k a1 s 1 1 t1 t ofthegroupg1 x g2 where pi aj resp p tj are nontrivial irreducible representations of g1 resp g2 we write g p v instead of g p v for simplicity to avoid confusion we write instead of clearly each gl 1 x g1 x g2 pi p i 1 k is an irreducible 2 simple p v if at least one of them is a nontrivial p v see definition 1 1 we call it a 2 simple p v of type i on the other hand if all of them are trivial p v s we call it a 2 simple p v of type ii in this case we have g2 sl n received by the editors march 3 1986 and in revised form july 30 1986 1980 mathematics subject classification 1985 revision primary 11r20 11r29 secondary 11n25 11r18 01988 american mathematical society 0002 9947 88 1 00 25 per page

poisson structures on affine spaces and flag varieties ii the standard poisson structures on the flag varieties g p of a complex reductive algebraic group g are investigated it is shown that the orbits of symplectic leaves in g p under a fixed maximal torus of g are smooth irreducible locally closed subvarieties of g p isomorphic to intersections of dual schubert cells in the full flag variety g b of g and their zariski closures are explicitly computed two different proofs of the former result are presented the first is in the framework of poisson homogeneous spaces and the second one uses an idea of weak splittings of surjective poisson submersions based on the notion of poisson dirac submanifolds for a parabolic subgroup p with abelian unipotent radical in which case g p is a hermitian symmetric space of compact type it is shown that all orbits of the standard levi factor l of p on g p are complete poisson subvarieties which are quotients of l equipped with the standard poisson structure moreover it is proved that the poisson structure on g p vanishes at all special base points for the l orbits on g p constructed by richardson rohrle and steinberg

the geometry of special symplectic representations abstract we show there is a class of symplectic lie algebra representations over any field of characteristic not 2 or 3 that have many of the exceptional algebraic and geometric properties of both symmetric three forms in two dimensions and alternating three forms in six dimensions the main algebraic result is that suitably generic elements of these representation spaces can be uniquely written as the sum of two elements of a naturally defined lagrangian subvariety we give universal explicit formulae for the summands and show how they lead to the existence of geometric structure on appropriate subsets of the representation space over the real numbers this structure reduces to either a conic special pseudo kahler metric or a conic special para kahler metric

quantum b functions of prehomogeneous vector spaces of commutative parabolic type we show that there exist natural q analogues of the b functions for the prehomogeneous vector spaces of commutative parabolic type and calculate them explicitly in each case our method of calculating the b functions seems to be new even for the original case q 1

q legendre transformation partition functions and quantization of the boltzmann constant in this paper we construct a q analogue of the legendre transformation where q is a matrix of formal variables defining the phase space braidings between the coordinates and momenta the extensive and intensive thermodynamic observables our approach is based on an analogy between the semiclassical wavefunctions in quantum mechanics and the quasithermodynamic partition functions in statistical physics the basic idea is to go from the q hamilton jacobi equation in mechanics to the q legendre transformation in thermodynamics it is shown that this requires a non commutative analogue of the planck boltzmann constants and kb to be introduced back into the classical formulae being applied to statistical physics this naturally leads to an idea to go further and to replace the quasithermodynamic parameter corresponding to the boltzmann constant with an infinite collection of generators of the so called epoche bracketing algebra the latter is an infinite dimensional non commutative algebra recently introduced in our previous work which can be perceived as an infinite sequence of deformations of deformations of the weyl algebra the generators mentioned are naturally indexed by planar binary leaf labelled trees in such a way that the trees with a single leaf correspond to the observables of the limiting thermodynamic system

representation and index theory for toeplitz operators we study toeplitz operators on the hardy spaces of connected compact abelian groups and of tube type bounded symmetric domains a representation theorem for these operators and for classes of abstract toeplitz elements in c algebras is proved this is used to give a unified treatment to index theory in this setting and a variety of new index theorems are proved that generalize the gohberg krein theorem for toeplitz operators on the hardy space of the unit circle in the plane

overall c as a measure of discrimination in survival analysis model specific population value and confidence interval estimation the assessment of the discrimination ability of a survival analysis model is a problem of considerable theoretical interest and important practical applications this issue is however more complex than evaluating the performance of a linear or logistic regression several different measures have been proposed in the biostatistical literature in this paper we investigate the properties of the overall c index introduced by harrell as a natural extension of the roc curve area to survival analysis we develop the overall c index as a parameter describing the performance of a given model applied to the population under consideration and discuss the statistic used as its sample estimate we discover a relationship between the overall c and the modified kendall s tau and construct a confidence interval for our measure based on the asymptotic normality of its estimate then we investigate via simulations the length and coverage probability of this interval finally we present a real life example evaluating the performance of a framingham heart study model

index theory for toeplitz operators on bounded symmetric domains in this note we give an index theory for toeplitz operators on the hardy space of the shilov boundary of an arbitrary bounded symmetric domain our results generalize earlier work of gohberg krein and venugopalkrishna 12 for domains of rank 1 and of berger coburn koranyi 1 for domains of rank 2 bounded symmetric domains cartan domains classical or exceptional are the natural higher dimensional analogues of the open unit disk each such domain is homogeneous under a semisimple lie group of biholomorphic transformations and has an essentially unique realization as a convex circular domain which is conveniently described in jordan algebraic terms the underlying vector space z c n carries a jordan triple product denoted by uyv w uv w and the bounded symmetric domain d is the open unit ball of z with respect to the spectral norm 7 the basic example is the space z c of rectangular matrices with jordan triple product uv w uv w wv u 2 giving rise to the hyperbolic matrix ball d z g z spectrum z z 1 the symmetric domains of rank 1 are the smooth hilbert balls in c n for domains d of higher rank r the boundary dd forms a nonsmooth stratified space in jordan algebraic terms the structure of dd can be described as follows an element e z satisfying ee e e is called a tripotent for matrices the tripotents are just the partial isometries every tripotent e induces a splitting of z into the peirce spaces z e z g z ee z xz for a 0 1 the set

a risk based measure of time varying prognostic discrimination for survival models prognostic survival models are commonly evaluated in terms of both their calibration and their discrimination comparing observed and predicted survival curves can assess calibration while discrimination is typically summarized through comparison of the properties of cases or subjects who experience an event and the properties of controls represented by event free individuals for binary data discrimination is characterized either by using the relative ranks of cases and controls and a receiver operating characteristic roc curve or by summarizing the magnitude of risk placed on cases and controls through calculation of the discrimination slope ds in this article we propose a risk based measure of time varying discrimination that generalizes the discrimination slope to allow use with incident events and hazard models we refer to the new measure as the hazard discrimination summary hds since it compares the relative risk among incident cases to their associated dynamic risk set controls we introduce both a model based estimation procedure that adopts the cox model and an alternative approach that locally relaxes the proportional hazards assumption we illustrate the proposed methods using both a benchmark survival data set and an oncology study where primary interest is in the time varying performance of candidate biomarkers

boundary orbit strata and faces of invariant cones and complex ol shanskiĭ semigroups let d g k be an irreducible hermitian symmetric domain then g is contained in a complexification gc and there exists a closed complex subsemigroup g gc characterised by fact that all holomorphic discrete series representations of g extend holomorphically to the socalled minimal ol shanskiuo semigroup parallel to the classical theory of boundary strata for the symmetric

non parabolic prehomogeneous vector spaces and exceptional lie algebras abstract among the classification of irreducible regular and reduced prehomogeneous vector spaces there are only a few one which are not of parabolic type in this paper we investigate this family of prehomogeneous vector and show that they nevertheless all occur inside the exceptional lie algebras and are in some sense close to the parabolic family moreover we give a lie theoretical description of their relative invariants and the corresponding characters