the hyperbolic class of quadratic time frequency representations i constant q warping the hyperbolic paradigm properties and members the time frequency tf version of the wavelet transform and the affine quadratic bilinear tf representations can be used for a tf analysis with constant q characteristic the paper considers a new approach to constant q tf analysis a specific tf warping transform is applied to cohen s class of quadratic tf representations which results in a new class of quadratic tf representations with constant q characteristic the new class is related to a hyperbolic tf geometry and is thus called the hyperbolic class hc two prominent tf representations previously considered in the literature the bertrand p sub 0 distribution and the altes marinovic q distribution are members of the new hc the authors show that any hyperbolic tf representation is related to both the wideband ambiguity function and a hyperbolic ambiguity function it is also shown that the hc is the class of all quadratic tf representations which are invariant to hyperbolic time shifts and tf scalings operations which are important in the analysis of doppler invariant signals and self similar random processes the paper discusses the definition of the hc via constant q warping some signal theoretic fundamentals of the hyperbolic tf geometry and the description of the hc by 2d kernel functions several members of the hc are considered and a list of desirable properties of hyperbolic tf representations is given together with the associated kernel constraints

a pieri type formula for isotropic flag manifolds we give the formula for multiplying a schubert class on an odd orthogonal or symplectic flag manifold by a special schubert class pulled back from the grassmannian of maximal isotropic subspaces this is also the formula for multiplying a type b respectively type c schubert polynomial by the schur p polynomial p m respectively the schur q polynomial q m geometric constructions and intermediate results allow us to ultimately deduce this formula from formulas for the classical flag manifold these intermediate results are concerned with the bruhat order of the infinite coxeter group b identities of the structure constants for the schubert basis of cohomology and intersections of schubert varieties we show that most of these identities follow from the pieri type formula and our analysis leads to a new partial order on the coxeter group b and formulas for many of these structure constants

the projective geometry of freudenthal s magic square abstract we connect the algebraic geometry and representation theory associated to freudenthal s magic square we give unified geometric descriptions of several classes of orbit closures describing their hyperplane sections and desingularizations and interpreting them in terms of composition algebras in particular we show how a class of invariant quartic polynomials can be viewed as generalizations of the classical discriminant of a cubic polynomial

grobner geometry of schubert polynomials given a permutation w sn we consider a determinantal ideal iw whose generators are certain minors in the generic n n matrix filled with independent variables using emultidegrees f as simple algebraic substitutes for torus equivariant cohomology classes on vector spaces our main theorems describe for each ideal iw

special prehomogeneous vector spaces associated to f4 e6 e7 e8 and simple jordan algebras of rank 3 abstract a simple complex lie algebra has a natural grading g g 2 g 1 g 0 g 1 g 2 associated to the highest root it is related to the quaternionic real form of g the real rank of the associated symmetric space is shown to be at most 4 when the rank is equal to 4 i e g a n c n g 2 the semi simple lie algebra m g 0 g 0 is shown to be the conformal algebra of a rank 3 semi simple jordan algebra v if g is one of the four exceptional lie algebras f4 e6 e7 e8 then v is simple the vector space g 1 is a prehomogeneous vector space under the action of m c where m int m is the adjoint group of m it admits a non zero m invariant polynomial of degree 4 conversely to any simple jordan algebra v of rank 3 is naturally associated a representation of a twofold covering of its conformal group co v on a vector space w such that w is a prehomogeneous vector space under the action of co v c an invariant polynomial of degree 4 is explicitly constructed a geometric description of all the orbits of co v in w is given

equivariant quantum schubert polynomials abstract we establish an equivariant quantum giambelli formula for partial flag varieties the answer is given in terms of a specialization of universal double schubert polynomials along the way we give new proofs of the presentation of the equivariant quantum cohomology ring as well as graham positivity of the structure constants in equivariant quantum schubert calculus

wigner ville spectral analysis of nonstationary processes the wigner ville spectrum has been recently introduced as the unique generalized spectrum for time varying spectral analysis its properties are revised with emphasis on its central role in the analysis of second order properties of nonstationary random signals we propose here a general class of spectral estimators of the wigner ville spectrum this class is based on arbitrarily weighted covariance estimators and its formal description corresponds to the general class of conjoint time frequency representations of deterministic signals with finite energy classical estimators like short time periodograms and the recently introduced pseudo wigner estimators are shown to be special cases of the general class the generalized framework allows the calculation of the moments of general spectral estimators and comparing the results emphasizes the versatility of the new pseudo wigner estimators the effective numerical implementation by an n point fft of pseudo wigner estimators of 2n points is indicated and various examples are given

time frequency formulation design and implementation of time varying optimal filters for signal estimation this paper presents a time frequency framework for optimal linear filters signal estimators in nonstationary environments we develop time frequency formulations for the optimal linear filter time varying wiener filter and the optimal linear time varying filter under a projection side constraint these time frequency formulations extend the simple and intuitive spectral representations that are valid in the stationary case to the practically important case of underspread nonstationary processes furthermore we propose an approximate time frequency design of both optimal filters and we present bounds that show that for underspread processes the time frequency designed filters are nearly optimal we also introduce extended filter design schemes using a weighted error criterion and we discuss an efficient time frequency implementation of optimal filters using multiwindow short time fourier transforms our theoretical results are illustrated by numerical simulations

divisors of secant planes to curves abstract inside the symmetric product of a very general curve we consider the codimension one subvarieties of symmetric tuples of points imposing exceptional secant conditions on linear series on the curve of fixed degree and dimension we compute the classes of such divisors and thus obtain improved bounds for the slope of the cone of effective divisor classes on symmetric products of a very general curve by letting the moduli of the curve vary we study more generally the classes of the related divisors inside the moduli space of stable pointed curves

f regularity of large schubert varieties let g denote a connected reductive algebraic group over an algebraically closed field k and let x denote a projective g g equivariant embedding of g the large schubert varieties in x are the closures of the double cosets bgb where b denotes a borel subgroup of g and g g we prove that these varieties are globally f regular in positive characteristic resp of globally f regular type in characteristic 0 as a consequence the large schubert varieties are normal and cohen macaulay 1 introduction the class of globally f regular varieties was introduced by smith in 21 these are projective algebraic varieties in positive character istics such that all the ideals in their homogeneous coordinate rings are tightly closed the globally f regular varieties and their analogues in characteristic 0 the varieties of globally f regular type have remarkable properties e g they are normal and cohen macaulay and the higher cohomology groups of all nef invertible sheaves are trivial examples of globally f regular varieties include the projective toric varieties prop 6 4 in 21 in this note we obtain the global f regularity of a wider class of varieties with algebraic group action the g g equivariant projective embeddings of any connected reductive group g and the closures in any such embedding of the double cosets bgb where b denotes a borel subgroup of g and g g is arbitrary by the bruhat decomposition these large schubert varieties are parametrized by the weyl group of g examples are the closures of parabolic subgroups we also show that the large schubert varieties are of globally f regular type in characteristic 0 for this we exploit the close relation between global f regularity and frobe nius splitting established in 21 and the frobenius splitting properties of the large schubert varieties proved in 5 and 19 another key ingredient is the global f regularity of the flag varieties and their schubert varieties 12 note that unlike for schubert varieties no desingularization of large schubert varieties is known in general this makes our arguments somewhat indirect as a consequence of our result the large schubert varieties in any equivariant embedding of g are normal and cohen macaulay this was first proved in the case of the canonical compactification of a semisimple adjoint group by frobenius

lagrangian subbundles and codimension 3 subcanonical subschemes we show that a gorenstein subcanonical codimension 3 subscheme z x p n 4 can be realized as the locus along which two lagrangian subbundles of a twisted orthogonal bundle meet degenerately and conversely we extend this result to singular z and all quasi projective ambient schemes x under the necessary hypothesis that z is strongly subcanonical in a sense defined below a central point is that a pair of lagrangian subbundles can be transformed locally into an alternating map in the local case our structure theorem reduces to that of d buchsbaum and d eisenbud 6 and says that z is pfaffian we also prove codimension 1 symmetric and skew symmetric analogues of our structure theorems 0 introduction smooth subvarieties of small codimension z x p have been extensively studied in recent years especially in relation to r hartshorne s conjecture that a smooth subvariety of sufficiently small codimension in p is a complete intersection although the conjecture remains open any smooth subvarietyz of small codimension in p is known by a theorem of w barth m larsen and s lefschetz to have the weaker property that it is subcanonical in the sense that its canonical class is a multiple of its hyperplane class more generally a subscheme z of a nonsingular noetherian scheme x is said to be subcanonical if z is gorenstein and its canonical bundle is the restriction of a bundle on x there is a natural generalization to an arbitrary possibly singular scheme x see below in this paper we give a structure theorem for subcanonical subschemes of codimension 3 in p and generalize it to subcanonical subschemes of codimension 3 in an arbitrary quasi projective scheme x satisfying a mild extra cohomological conduke mathematical journal vol 107 no 3 c 2001 received 4 july 1999 revision received 17 july 2000 2000 mathematics subject classification primary 14m07 13d02 secondary 14m12 14f05 authors work partially supported by the national science foundation

magic squares and matrix models of lie algebras this paper is concerned with the description of exceptional simple lie algebras as octonionic analogues of the classical matrix lie algebras we review the tits freudenthal construction of the magic square which includes the exceptional lie algebras as the octonionic case of a construction in terms of a jordan algebra of hermitian 3 3 matrices tits or various plane and other geometries freudenthal we present alternative constructions of the magic square which explain its symmetry and show explicitly how the use of split composition algebras leads to analogues of the matrix lie algebras su 3 sl 3 and sp 6 we adapt the magic square construction to include analogues of su 2 sl 2 and sp 4 for all real division algebras

a time frequency formulation of optimum detection a time frequency formulation is proposed for the optimum detection of gaussian signals in white gaussian noise by choosing the wigner ville distribution as the basic time frequency tool it is shown that the corresponding receivers generally take the form of a correlation between time frequency structures matching mathematical optimality with a physically meaningful interpretation the case of low snr is examined and various examples are considered deterministic signal rayleigh fading signal random jitter and random time varying channel a general class of time frequency receivers is proposed which admits as limiting cases different known structures and its suboptimum performance is evaluated possible extensions to more elaborate situations including parameter estimation are mentioned

presenting the cohomology of a schubert variety we extend the short presentation due to borel 53 of the cohomology ring of a generalized flag manifold to a relatively short presentation of the cohomology of any of its schubert varieties our result is stated in a root system uniform manner by introducing the essential set of a coxeter group element generalizing and giving a new characterization of fulton 92 s definition for permutations further refinements are obtained in type a

the weyl correspondence and time frequency analysis describes the weyl correspondence and its properties showing how it gives a window independent definition of time frequency concentration for use in models in signal detection the definition of concentration is justified by showing that it gives reasonable answers in certain intuitive cases the weyl correspondence expresses a linear transformation as a weighted superposition of time frequency shifts of the signal and then authors explain why this is not the same as transforming a signal into the time frequency domain multiplying by a weight in the transform domain and taking the inverse the investigation into time frequency concentration and the weyl correspondence is justified by a new result the authors show that convolving the wigner distribution with a general smoothing function is equivalent to evaluating a weighted sum of spectrograms this is a new interpretation of the process of smoothing the wigner distribution to reduce cross terms it relates smoothing of the wigner distribution to the multiple window technique pioneered by thomson 1982

asymptotic behavior of matrix coefficients of admissible representations introduction 869 1 generalities on reductive groups 873 2 the infinitesimal cartan decomposition 876 3 the t radial components 880 4 differential equations satisfied by sphereical functions 885 5 asymptotic behavior of spherical functions on a 888 6 asymptotic behavior of spherical functions along the walls of a 892 7 leading characters and growth estimates on the group 898 8 admissible representations and their matrix coefficients 906 appendix 917

schubert polynomials and classes of hessenberg varieties abstract regular semisimple hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory numerical analysis representation theory algebraic geometry and combinatorics we give a giambelli formula expressing the classes of regular semisimple hessenberg varieties in terms of chern classes in fact we show that the cohomology class of each regular semisimple hessenberg variety is the specialization of a certain double schubert polynomial giving a natural geometric interpretation to such specializations we also decompose such classes in terms of the schubert basis for the cohomology ring of the flag variety the coefficients obtained are nonnegative and we give closed combinatorial formulas for the coefficients in many cases we introduce a closely related family of schemes called regular nilpotent hessenberg schemes and use our results to determine when such schemes are reduced

the sextonions and e712 we fill in the hole in the exceptional series of lie algebras that was observed by cvitanovic deligne cohen and deman more precisely we show that the intermediate lie algebra between e7 and e8 satisfies some of the decomposition and dimension formulas of the exceptional simple lie algebras a key role is played by the sextonions a six dimensional algebra between the quaternions and octonions using the sextonions we show similar results hold for the rows of an expanded freudenthal magic chart we also obtain new interpretations of the adjoint variety of the exceptional group g2

on the transfer function calculus for underspread ltv channels the freedom in the definition of a time varying transfer function is formalized by introducing the generalized weyl correspondence which includes zadeh s 1950 function and the weyl symbol for slowly time varying linear systems with properly limited memory underspread ltv systems we prove the asymptotic validity of the multiplicative transfer function calculus

ein neuer zusammenhang zwischen einfachen gruppen und einfachen singularitaten we describe a new construction to obtain a simple hypersurface singularity from the corresponding simple complex lie groupg letx be the closed orbit in the projective space attached to the lie algebra mathfrak g ofg consider a regular nilpotent element y 0 in mathfrak g and denote byh y 0 the hyperplane orthogonal toy 0 with respect to the killing form then the hyperplane sectionx h y 0 has exactly one singularity which is simple of desired type by variation of the pointy 0 we obtain a versal deformation the construction generalizes with minor modifications to any characteristicp of the basefield even in bad characteristic we recover at least the positive part of the semiuniversal deformation we prove that forp 2 a simple quasihomogeneous singularity of type a7 resp d8 is adjacent to e7 resp e8 provided its dimension is even furthermore a8 is adjacent to e8 forp 3

representations of reductive groups distinguished by symmetric subgroups let g be a complex connected reductive group g theta be its fixed point subgroup under a galois involution theta and h be an open subgroup of g theta we show that any h distinguished representation pi satisfies 1 pi theta simeq tilde pi where tilde pi is the contragredient representation and pi theta is the twist of pi under theta 2 mathrm dim mathbb c left pi right h le left b backslash g h right where b is a borel subgroup of g by proving the first statement we give a partial answer to a conjecture by prasad and lapid

modular linear differential equations of fourth order and minimal w algebras abstract a characterization of the minimal w algebras associated with the deligne exceptional series at level h 6 is obtained by using one parameter family of modular linear differential equations of order 4 in particular the characters of the ramond twisted modules of minimal w algebras related to the deligne exceptional series satisfy one of these differential equations in order to obtain the characterization the differential equations in the one parameter family which have solutions of cft type are classified whose solutions are explicitly described

subword complexes in coxeter groups abstract let be a coxeter system an ordered list of elements in and an element in determine a subword complex as introduced in knutson and miller ann of math 2 2003 to appear subword complexes are demonstrated here to be homeomorphic to balls or spheres and their hilbert series are shown to reflect combinatorial properties of reduced expressions in coxeter groups two formulae for double grothendieck polynomials one of which appeared in fomin and kirillov proceedings of the sixth conference in formal power series and algebraic combinatorics dimacs 1994 pp 183 190 are recovered in the context of simplicial topology for subword complexes some open questions related to subword complexes are presented

the power classes quadratic time frequency representations with scale covariance and dispersive time shift covariance we consider scale covariant quadratic time frequency representations qtfrs specifically suited for the analysis of signals passing through dispersive systems these qtfrs satisfy a scale covariance property that is equal to the scale covariance property satisfied by the continuous wavelet transform and a covariance property with respect to generalized time shifts we derive an existence representation theorem that shows the exceptional role of time shifts corresponding to group delay functions that are proportional to powers of frequency this motivates the definition of the power classes pcs of qtfr s the pcs contain the affine qtfr class as a special case and thus they extend the affine class we show that the pcs can be defined axiomatically by the two covariance properties they satisfy or they can be obtained from the affine class through a warping transformation we discuss signal transformations related to the pcs the description of the pcs by kernel functions desirable properties and kernel constraints and specific pc members furthermore we consider three important pc subclasses one of which contains the bertrand 1992 p sub k distributions finally we comment on the discrete time implementation of pc qtfrs and we present simulation results that demonstrate the potential advantage of pc qtfrs

finite multiplicity theorems for induction and restriction abstract we find upper and lower bounds of the multiplicities of irreducible admissible representations of a semisimple lie group g occurring in the induced representations ind h g from irreducible representations of a closed subgroup h as corollaries we establish geometric criteria for finiteness of the dimension of hom g ind h g induction and of hom h h restriction by means of the real flag variety g p and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety

double schubert polynomials for the classical groups abstract for each infinite series of the classical lie groups of type b c or d we construct a family of polynomials parametrized by the elements of the corresponding weyl group of infinite rank these polynomials represent the schubert classes in the equivariant cohomology of the appropriate flag variety they satisfy a stability property and are a natural extension of the single schubert polynomials of billey and haiman which represent non equivariant schubert classes they are also positive in a certain sense and when indexed by maximal grassmannian elements or by the longest element in a finite weyl group these polynomials can be expressed in terms of the factorial analogues of schur s q or p functions defined earlier by ivanov

chern schwartz macpherson classes and the euler characteristic of degeneracy loci and special divisors this concept overlaps a large family of interesting varieties for example the varieties of special divisors studied in section 3 several authors have worked out explicit formulas for the euler characteristic of dr 9 in terms of different cohomological and numerical invariants under the assumption that x is nonsingular and v is appropriately general for instance if 0 is a section of a vector bundle then the formulas for the euler characteristic x do fp were given by hirzebruch h and navarro aznar n if dr is a curve or a surface in a nonsingular x some explicit formulas for x dr v were given by harris and tu h t in terms of the chern classes of e f and x but under the extra assumption dr 1 v 0 which implies that dr q is nonsingular in loc cit the authors also posed the problem of finding a general formula for x dr o if such exists under the assumption dr 1 0 or even stronger without this assumption the first problem was solved positively by the second named author prl proposition 5 7 by the use of polynomials universally supported on degeneracy

wide band proportional bandwidth wigner ville analysis the wigner ville w v distribution is a time frequency representation that yields a highly accurate estimate of instantaneous frequency it is related to the narrow band ambiguity function by an internal transform and it can be used in a variety of detection and estimation problems a spectrogram constructed with constant bandwidth filters can be obtained by convolving two w v distributions or by forming a magnitude squared narrow band cross ambiguity function the wide band ambiguity function represents the doppler effect with dilation or compression rather than with frequency shift as in the narrow band approximation the q distributions a modified w v representation that is related to the wide band ambiguity function by an integral transform is defined a spectrogram constructed with proportional bandwidth or constant q filters can be obtained by a convolution like operation involving two q distributions or by forming a magnitude squared wide band cross ambiguity function the q distribution is thus a wide band version of the w v distribution properties of the q distribution indicate that it may prove useful for detection and parameter estimation as well as for tomographic measurement of wideband scattering functions with relatively few transmitted waveforms

triality exceptional lie algebras and deligne dimension formulas abstract we give a computer free proof of the deligne cohen and de man formulas for the dimensions of the irreducible g modules appearing in g k k 4 where g ranges over the exceptional complex simple lie algebras we give additional dimension formulas for the exceptional series as well as uniform dimension formulas for other representations distinguished by freudenthal along the rows of his magic chart our proofs use the triality model of the magic square which we review and present a simplified proof of its validity we conclude with some general remarks about obtaining series of lie algebras in the spirit of deligne and vogel

generalized evolutionary spectral analysis and the weyl spectrum of nonstationary random processes the evolutionary spectrum es is a time varying power spectrum of nonstationary random processes starting from an innovations system interpretation of the es we introduce the generalized evolutionary spectrum ges as a novel family of time varying power spectra the ges contains the es and the transitory evolutionary spectrum as special cases we consider the problem of finding an innovations system for a process characterized by its correlation function and we discuss the connection between ges analysis and the class of underspread processes furthermore we show that another special case of the ges a novel time varying power spectrum that we call the weyl spectrum has substantial advantages over all other members of the ges family the properties of the weyl spectrum are discussed and its superior performance is verified experimentally for synthetic and real data processes

characteristic classes and quadric bundles in this paper we construct stiefel whitney and euler classes in chow cohomology for algebraic vector bundles with nondegenerate quadratic form these classes are not in the algebra generated by the chern classes of such bundles and are new characteristic classes in algebraic geometry on complex varieties they correspond to classes with the same name pulled back from the cohomology of the classifying space bso n c the classes we construct are the only new characteristic classes in algebraic geometry coming from the classical groups t2 eg we begin by using the geometry of quadric bundles to study chern classes of maximal isotropic subbundles if v x is a vector bundle with quadratic form and if e and f are maximal isotropic subbundles of v then we prove theorem 1 that ci e and ci f are equal mod 2 moreover if the rank of v is 2n then cn e cn f proving a conjecture of fulton we define stiefel whitney and euler classes as chow cohomology classes which pull back to chern classes of maximal isotropic subbundles of the pullback bundle using the above theorem we show theorem 2 that these classes exist and are unique even though v need not have a maximal isotropic subbundle these constructions also make it possible to give schubert presentations

time frequency transfer function calculus symbolic calculus of linear time varying systems linear operators based on a generalized underspread theory we introduce a generalized concept of underspread linear time varying systems linear operators which contains a previous definition as a special case we show that an existing approximate transfer function calculus symbolic calculus can be extended to this wider and practically more relevant class of underspread operators as a mathematical underpinning of this calculus we establish explicit bounds on various error quantities associated with it the transfer function calculus provides a theoretical basis for various methods recently proposed for nonstationary signal processing and it has important implications in the theory of time varying power spectra

composition algebras over algebraic curves of genus zero we rephrase the classical theory of composition algebras over fields particularly the cayley dickson doubling process and zorn s vector matrices in the setting of locally ringed spaces fixing an arbitrary base field we use these constructions to classify composition algebras over complete smooth curves of genus zero applications are given to composition algebras over function fields of genus zero and polynomial rings

odd symplectic groups on definit des groupes symplectiques sur des espaces vectoriels de dimension impaire ces groupes de lie ne sont ni simples ni reductifs on obtient une formule de caracteres pour certaines representations tensorielles de ces groupes

group delay shift covariant quadratic time frequency representations we propose classes of quadratic time frequency representations qtfrs that are covariant to group delay shifts gdss the gds covariance qtfr property is important for analyzing signals propagating through dispersive systems with frequency dependent characteristics this is because a qtfr satisfying this property provides a succinct representation whenever the time shift is selected to match the frequency dependent changes in the signal s group delay that may occur in dispersive systems we obtain the gds covariant classes from known qtfr classes such as cohen s 1995 class the affine class the hyperbolic class and the power classes using warping transformations that depend on the relevant group delay change we provide the formulation of the gds covariant classes using two dimensional 2 d kernel functions and we list desirable qtfr properties and kernel constraints as well as specific class members we present known examples of the gds covariant classes and we provide a new class the power exponential qtfr class we study the localized kernel subclasses of the gds covariant classes that simplify the theoretical development as the kernels reduce from 2 d to one dimensional 1 d functions and we develop various intersections between the qtfr classes finally we present simulation results to demonstrate the advantage of using qtfrs that are matched to changes in the group delay of a signal

unitary equivalence a new twist on signal processing unitary similarity transformations furnish a powerful vehicle for generating infinite generic classes of signal analysis and processing tools based on concepts different from time frequency and scale implementation of these new tools involves simply preprocessing the signal by a unitary transformation performing standard processing on the transformed signal and then in some cases transforming the resulting output the resulting unitarily equivalent systems can focus on the critical signal characteristics in large classes of signals and hence prove useful for representing and processing signals that are not well matched by current techniques as specific examples of this procedure we generalize linear time invariant systems orthonormal basis and frame decompositions and joint time frequency and time scale distributions these applications illustrate the utility of the unitary equivalence concept for uniting seemingly disparate approaches proposed in the literature

the hyperbolic class of quadratic time frequency representations ii subclasses intersection with the affine and power classes regularity and unitarity for pt i see ibid vol 41 p 3425 444 1993 part i introduced the hyperbolic class hc of quadratic bilinear time frequency representations qtfrs the present paper defines and studies four subclasses of the hc 1 the focalized kernel subclass of the hc is related to a time frequency concentration property of qtfrs it is analogous to the localized kernel subclass of the affine qtfr class 2 the affine subclass of the hc affine hc consists of all hc qtfrs that satisfy the conventional time shift covariance property it forms the intersection of the hc with the affine qtfr class 3 the power subclasses of the hc consist of all hc qtfrs that satisfy a power time shift covariance property they form the intersection of the hc with the recently introduced power classes 4 the power warp subclass of the hc consists of all hc qtfrs that satisfy a covariance to power law frequency warpings it is the hc counterpart of the shift scale covariant subclass of cohen s class all of these subclasses are characterized by 1d kernel functions the affine hc is contained in both the localized kernel hyperbolic subclass and the localized kernel affine subclass and that any affine hc qtfr can be derived from the bertrand unitary po distribution by a convolution we furthermore consider the properties of regularity and unitarity in the hc the calculus of inverse kernels is developed and important implications of regularity and unitarity are summarized the results comprise a general method for least squares signal synthesis and new relations for the altes marinovich q distribution

symbolic calculus on the time frequency half plane this study concerns a special symbolic calculus of interest for signal analysis this calculus associates functions on the time frequency half plane f 0 with linear operators defined on the positive frequency signals full attention is given to its construction which is entirely based on the study of the affine group in a simple and direct way the correspondence rule is detailed and the associated wigner function is given formulas expressing the basic operation star bracket of the lie algebra of symbols which is isomorphic to that of the operators are obtained in addition it is shown that the resulting calculus is covariant under a three parameter group which contains the affine group as subgroup this observation is the starting point of an investigation leading to a whole class of symbolic calculi which can be considered as modifications of the original one

regularity and unitarity of bilinear time frequency signal representations two structural properties of bilinear time frequency representations btfrs of signals are introduced and studied the definition of these properties is based on a linear operator description of btfrs the first property termed regularity has important implications with respect to the recovery of signals from the btfr outcome the derivation of other bilinear signal representations from a btfr the btfrs reaction to linear signal transformations and the construction of bases of induced btfr domain spaces the second property called unitarity is equivalent to validity of moyal s formula 1949 unitarity is thus necessary and sufficient for a closed form solution of optimal signal synthesis and for a btfr formulation or optimal detection estimation methods unitarity also allows the systematic construction of btfr product relations like the wigner distribution s interference formula and the ambiguity function s self transformation property unitarity permits the construction of induced orthogonal projection operators and guarantees the orthonormality of induced basis functions

kostant polynomials and the cohomology ring for g b the schubert calculus for g b can be completely determined by a certain matrix related to the kostant polynomials introduced in section 5 of bernstein gelfand and gelfand bernstein i gelfand i gelfand s 1973 russ math surv 28 1 26 the polynomials are defined by vanishing properties on the orbit of a regular point under the action of the weyl group for each element w in the weyl group the polynomials also have nonzero values on the orbit points corresponding to elements which are larger in the bruhat order than w the main theorem given here is an explicit formula for these values the matrix of orbit values can be used to determine the cup product for the cohomology ring for g b using only linear algebra or as described by lascoux and schutzenberger lascoux a schutzenberger m p 1982 c r seances acad sci ser a 294 447 450 complete proofs of all the theorems will appear in a forthcoming paper

wideband weyl symbols for dispersive time varying processing of systems and random signals we extend the narrowband weyl symbol ws and the wideband p sub o weyl symbol pows for dispersive time frequency tf analysis of nonstationary random processes and time varying systems we obtain the new tf symbols using unitary transformations on the ws and the p sub o ws for example whereas the ws is matched to systems with constant or linear tf characteristics the new symbols are better matched to systems with dispersive nonlinear tf structures this results from matching the geometry of the unitary transformation to the specific tf characteristics of a system we also develop new classes of smoothed weyl symbols that are covariant to tf shifts or time shift and scaling system transformations these classes of symbols are also extended via unitary warpings to obtain classes of tf symbols covariant to dispersive shifts we provide examples of the new symbols and symbol classes and we list some of their desirable properties using simulation examples we demonstrate the advantage of using tf symbols that are matched to the changes in the tf characteristics of a system or random process we also provide new tf formulations for matched detection applications

volume growth temperedness and integrability of matrix coefficients on a real spherical space abstract we apply the local structure theorem from 13 and the polar decomposition of 12 to a real spherical space z g h and control the volume growth on z we define the harish chandra schwartz space on z we give a geometric criterion to ensure l p integrability of matrix coefficients on z

metrics with exceptional holonomy it is proved that there exist metrics with holonomy g2 and spin 7 thus settling the remaining cases in berger s list of possible holonomy groups we first reformulate the holonomy h condition as a set of differential equations for an associated h structure on a given manifold we collect the needed algebraic facts about g2 and spin 7 we then apply the machinery of over determined partial differential equations in the form of the cartan kahler theorem to prove the existence of solutions whose holonomy is g2 or spin 7 we also provide explicit examples and some information about the generality of the space of

a scheme related to the brauer loop model abstract we introduce the brauer loop scheme e m m n c m m 0 where is a certain degeneration of the ordinary matrix product its components of top dimension n 2 2 correspond to involutions s n having one or no fixed points in the case n even this scheme contains the upper upper scheme from a knutson some schemes related to the commuting variety j algebraic geom in press math ag 0306275 as a union of n 2 of its components one of those is a degeneration of the commuting variety of pairs of commuting matrices the brauer loop model is an integrable stochastic process studied in j de gier b nienhuis brauer loops and the commuting variety j stat mech 2005 p01006 math ag 0410392 based on earlier related work in m j martins b nienhuis r rietman an intersecting loop model as a solvable super spin chain phys rev lett 81 1998 504 507 cond mat 9709051 and some of the entries of its perron frobenius eigenvector were observed conjecturally to equal the degrees of the components of the upper upper scheme our proof of this equality follows the program outlined in p di francesco p zinn justin inhomogeneous model of crossing loops and multidegrees of some algebraic varieties math ph 0412031 in that paper the entries of the perron frobenius eigenvector were generalized from numbers to polynomials which allowed them to be calculated inductively using divided difference operators we relate these polynomials to the multi degrees of the components of the brauer loop scheme defined using an evident torus action on e as a consequence we obtain a formula for the degree of the commuting variety previously calculated up to 4 4 matrices

duality and classification of bilinear time frequency signal representations the author discusses a fundamental duality principle of btfrs bilinear time frequency representations and presents a systematic classification of btfrs which is consistent with btfr duality btfrs are first grouped into two basic domains namely the energy density domain e domain with energetic interpretation and the correlation domain c domain with correlative interpretation it is shown that these domains are related by a fourier transform duality to any btfr btfr relation or btfr property of the e domain there corresponds a dual btfr btfr relation or btfr properties of the c domain and vice versa with this duality principle as a background a classification of btfrs is given this classification is based on two dual shift invariance properties and a self dual scale invariance property of btfrs the mathematical description of btfrs by means of kernel functions is simplified inside the respective btfr classes it is shown that btfrs which are both shift and scale invariant can be represented as superpositions of generalized wigner distributions e domain or generalized ambiguity function c domain

a tableau formula for eta polynomials we use the pieri and giambelli formulas of buch et al invent math 178 345 405 2009 j reine angew 2013 and the calculus of raising operators developed in buch et al a giambelli formula for isotropic grassmannians arxiv 0811 2781 2008 and tamvakis j reine angew math 652 207 244 2011 to prove a tableau formula for the eta polynomials of buch et al j reine angew 2013 and the stanley symmetric functions which correspond to grassmannian elements of the weyl group widetilde w n of type text d n we define the skew elements of widetilde w n and exhibit a bijection between the set of reduced words for any skew w in widetilde w n and a set of certain standard typed tableaux on a skew shape lambda mu associated to w

erratum to hausdorffness for lie algebra homology of schwartz spaces and applications to the comparison conjecture let h be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold x and a real algebraic bundle mathcal e on x let mathfrak h be the lie algebra of h let mathcal s x mathcal e be the space of schwartz sections of mathcal e we prove that mathfrak h mathcal s x mathcal e is a closed subspace of mathcal s x mathcal e of finite codimension we give an application of this result in the case when h is a real spherical subgroup of a real reductive group g we deduce an equivalence of two old conjectures due to casselman the automatic continuity and the comparison conjecture for zero homology namely let pi be a casselman wallach representation of g and v be the corresponding harish chandra module then the natural morphism of coinvariants v mathfrak h rightarrow pi mathfrak h is an isomorphism if and only if any linear mathfrak h invariant functional on v is continuous in the topology induced from pi the latter statement is known to hold in two important special cases if h includes a symmetric subgroup and if h includes the nilradical of a minimal parabolic subgroup of g

a class of affine wigner functions with extended covariance properties affine wigner functions are phase space representations based on the affine group in place of the usual weyl heisenberg group of quantum mechanics such representations are relevant to the time frequency analysis of real signals an interesting family is singled out by the requirement of covariance with respect to each solvable three parameter group containing the affine group explicit forms are given in each case and properties such as unitarity and localization are discussed some particular distributions are recovered

the cayley grassmannian abstract we study the projective variety cg parametrizing four dimensional subalgebras of the complex octonions which we call the cayley grassmannian we prove that it is a spherical g 2 variety with only three orbits that we describe explicitly its cohomology ring has a basis of schubert type classes and we determine the intersection product completely

a universal dimension formula for complex simple lie algebras we present a universal formula for the dimension of the cartan powers of the adjoint representation of a complex simple lie algebra i e a universal formula for the hilbert functions of homogeneous complex contact manifolds as well as several other universal formulas these formulas generalize formulas of vogel and deligne and are given in terms of rational functions where both the numerator and denominator decompose into products of linear factors with integer coefficients we discuss consequences of the formulas including a relation with scorza varieties

exceptional lie algebras su 3 and jordan pairs part 2 zorn type representations a representation of the exceptional lie algebras reflecting a simple unifying view based on realizations in terms of zorn type matrices is presented the role of the underlying jordan pair and jordan algebra content is crucial in the development of the structure each algebra contains three jordan pairs sharing the same lie algebra of automorphisms and the same external su 3 symmetry the applications in physics are outlined

a directed graph structure of alternating sign matrices abstract we introduce a new directed graph structure into the set of alternating sign matrices this includes bruhat graph bruhat order of the symmetric groups as a subgraph subposet drake gerrish skandera 2004 2006 6 7 gave characterizations of bruhat order in terms of total nonnegativity tnn and subtraction free laurent sfl expressions for permutation monomials with our directed graph we extend their idea in two ways first from permutations to alternating sign matrices second q analogs which we name q tnn and q sfl properties as a by product we obtain a new kind of permutation statistic the signed bigrassmannian statistics using dodgson s condensation on determinants

time scale energy distributions a general class extending wavelet transforms the theory of a new general class of signal energy representations depending on time and scale is developed time scale analysis has been introduced recently as a powerful tool through linear representations called continuous wavelet transforms wts a concept for which an exhaustive bilinear generalization is given although time scale is presented as an alternative method to time frequency strong links relating the two are emphasized thus combining both descriptions into a unified perspective the authors provide a full characterization of the new class the result is expressed as an affine smoothing of the wigner ville distribution on which interesting properties may be further imposed through proper choices of the smoothing function parameters not only do specific choices allow recovery of known definitions but they also provide via separable smoothing a continuous transition from wigner ville to either spectrograms or scalograms squared modulus of the wt this property makes time scale representations a very flexible tool for nonstationary signal analysis

a grobner basis for kazhdan lusztig ideals it kazhdan lusztig ideals a family of generalized determinantal ideals investigated in woo yong 08 provide an explicit choice of coordinates and equations encoding a neighborhood of a torus fixed point of a schubert variety on a type a flag variety our main result is a gr o bner basis for these ideals this provides a single geometric setting to transparently explain the naturality of pipe dreams on the it rothe diagram of a permutation and their appearance in begin itemize item combinatorial formulas fomin kirillov 94 for schubert and grothendieck polynomialsof lascoux sch u tzenberger 82 item the equivariant k theory specialization formula of buch rim a nyi 04 and item a positive combinatorial formula for multiplicities of schubert varieties in good cases including those for which the associated kazhdan lusztig ideal is homogeneous under the standard grading end itemize our results generalize with alternate proofs knutson miller 05 s gr o bner basis theorem for schubert determinantal ideals and their geometric interpretation of the monomial positivity of schubert polynomials we also complement recent work of knutson 08 09 on degenerations of kazhdan lusztig varieties in general lie type as well as work of goldin 01 on equivariant localization and of lakshmibai weyman 90 rosenthal zelevinsky 01 and krattenthaler 01 on grassmannian multiplicity formulas

when is a schubert variety gorenstein abstract a normal variety is gorenstein if it is cohen macaulay and its canonical sheaf is a line bundle this property is a weakening of smoothness which implies that a variety behaves like a smooth one for various algebro geometric purposes we give a combinatorial characterization of which schubert varieties are gorenstein using in part a new generalization of pattern avoidance we also give an explicit description as a line bundle of the canonical sheaf of a gorenstein schubert variety

schubert polynomials for the classical groups the task of a theory of schubert polynomials is to produce explicit representatives for schubert classes in the cohomology ring of a flag variety and to do so in a manner that is as natural as possible from a combinatorial point of view to explain more fully let us review a special case the schubert calculus for grassmannians where one asks for the number of linear spaces of given dimension satisfying certain geometric conditions a typical problem is to find the number of lines meeting four given lines in general position in 3 space answer below for each of the four given lines the set of lines meeting it is a schubert variety in the grassmannian and we want the number of intersection points of these four subvarieties in the modem solution of this problem the schubert varieties induce canonical elements of the cohomology ring of the grassmannian called schubert classes the product of these schubert classes is the class of a point times the number of intersection points counted with appropriate multiplicities this reformulation of the problem though one of the great achievements of algebraic geometry is only part of a solution it remains to give a concrete model for the cohomology ring that makes explicit computation with schubert classes possible as it happens the cohomology rings of grassmannians can be identified with quotients of a polynomial ring so that schubert classes correspond to schur functions intersection numbers such as we are considering then turn out to be littlewood richardson coefficients for example the answer to our four lines 4 problem is the coefficient of the schur function s 2 2 in the product s 1 x or 2 for an extended treatment and history of the subject see 10 11 18 the identification of schur functions as schubert polynomials for grassmannians is a consequence of a more general and now highly developed theory of schubert polynomials for the flag varieties of the special linear groups sl n c the starting point for this more general theory is a construction of

structurable algebras and groups of type e6 and e7 abstract it is well known that every group of type f 4 is the automorphism group of an exceptional jordan algebra and that up to isogeny all groups of type 1 e 6 with trivial tits algebras arise as the isometry groups of norm forms of such jordan algebras we describe a similar relationship between groups of type e 6 and groups of type e 7 and use it to give explicit descriptions of the homogeneous projective varieties associated to groups of type e 7 with trivial tits algebras

albert algebras over curves of genus zero and one abstract albert algebras and other jordan algebras are constructed over curves of genus zero and one using a generalization of the tits process and the first tits construction due to achhammer

some degenerations of kazhdan lusztig ideals and multiplicities of schubert varieties abstract we study hilbert samuel multiplicity for points of schubert varieties in the complete flag variety by grobner degenerations of the kazhdan lusztig ideal in the covexillary case we give a manifestly positive combinatorial rule for multiplicity by establishing with a grobner basis a reduced limit whose stanley reisner simplicial complex is homeomorphic to a shellable ball or sphere we show that multiplicity counts the number of facets of this complex we also obtain a formula for the hilbert series of the local ring in particular our work gives a multiplicity rule for grassmannian schubert varieties providing alternative statements and proofs to formulae of lakshmibai and weyman 1990 26 rosenthal and zelevinsky 2001 37 krattenthaler 2001 22 kodiyalam and raghavan 2003 21 kreiman and lakshmibai 2004 24 ikeda and naruse 2009 13 and woo and yong 2009 40 we suggest extensions of our methodology to the general case

decay of matrix coefficients on reductive homogeneous spaces of spherical type let z be a homogeneous space z g h of a real reductive lie group g with a reductive subgroup h the investigation concerns the quantitative decay of matrix coefficients on z under the assumption that z is of spherical type that is minimal parabolic subgroups have open orbits on z

the nash blow up of a cominuscule schubert variety abstract we compute the nash blow up of a cominuscule schubert variety in particular we show that the nash blow up is algebraically isomorphic to another schubert variety of the same lie type as a consequence we give a new characterization of the smooth locus and for grassmannian schubert varieties determine when the nash blow up is a resolution of singularities we also study the induced torus action on the nash blow up and give a bijection between its torus fixed points and peterson translates on the schubert variety