boson realization of sp 4 r ii the generating kernel formulation in a previous paper of this series the matrix elements were discussed with respect to boson states of an operator k2 required for the boson realization of the sp 4 r lie algebra in the present paper it is shown that these matrix elements can be obtained from a generating kernel given by the overlap of sp 4 r coherent states the results have relevance for the determination of the matrix elements of the generators of the sp 4 r lie algebra with respect to the basis of irreps of the positive discrete series for the corresponding group and are in principle generalizable to symplectic algebras of higher dimensions

canonical orthonormal wigner supermultiplet basis the explicit construction of an orthonormal basis for states of good spin isospin and su 4 wigner supermultiplet symmetry is given in a bargmann representation space a complete set of quantum labels is provided by a sp 3 r contains implies u 3 complementary symmetry

weak type 1 1 of some operators for the laplacian with drift let v v 1 ldots v n be a vector in mathbb r n setminus 0 consider the laplacian on mathbb r n with drift delta v sum i 1 n big frac partial 2 partial x i 2 2 v i frac partial partial x i big and the measure d mu x e 2 langle v x rangle dx with respect to which delta v is self adjoint let d and nabla denote the euclidean distance and the gradient operator on mathbb r n consider the space mathbb r n d d mu which has the property of exponential volume growth we obtain weak type 1 1 for the riesz transform nabla delta v frac 1 2 and for the heat maximal operator with respect to d mu further we prove that the uncentered hardy littlewood maximal operator is bounded on l p for 1 p le infty but not of weak type 1 1 if n ge 2

sliding window adaptive fast qr and qr lattice algorithms sliding window formulations of the fast qr and fast qr lattice algorithms are presented the derivations are based on the partial triangularization of raw data matrices three methods for window downdating are discussed the method of plane hyperbolic rotations the chambers method and the linpack algorithm a numerically ill conditioned stationary signal and a speech signal are used in finite wordlength simulations of the full qr nonfast fast qr and qr lattice algorithms all algorithms are observed to be numerically stable over billions of iterations for double precision mantissas 53 bits but as the number of bits is decreased in the mantissa the algorithms exhibit divergent behavior hence practically the algorithms can de regarded as numerically stable for long wordlengths

lp estimates on functions of the laplace operator on etudie la continuite l p de certaines fonctions du laplacien sur une variete de riemann complete a geometrie bornee

internal labelling problem an algorithmic procedure combining the decomposition of casimir operators induced by the embedding of a subalgebra into a semisimple lie algebra with the properties of commutators of subgroup scalars an analytical algorithm for the computation of missing label operators with the commutativity requirement is proposed two new criteria for subgroups scalars to commute are given the algorithm is completed with a recursive method to construct orthonormal bases of states as examples to illustrate the procedure four labelling problems are explicitly studied

analytic expressions for the matrix elements of generators of sp 6 in an sp 6 u 3 basis work done by many authors indicates that important tools for calculations in microscopic collective models are the matrix elements of the generators of the symplectic group sp 6 in an sp 6 u 3 basis rosensteel has derived recursion relations for these matrix elements while filippov has determined them using generating function technique but it would also be convenient to have explicit and analytic formulas for them this is what we do in this paper for the case of closed shells i e when the irreducible representation irrep of sp 6 is characterized by equal values for the three weight generators in the lowest weight state we also indicate how our results can be extended to the case of arbitrary irreps of sp 6 i e when we have open shells

resolution of missing label problems a new perspective on k matrix theory k matrix theory is an effective algebraic way of determining inner products and constructing orthonormal bases vectors for the carrier spaces of induced representations of lie groups and lie algebras the theory is applied in this paper to give canonical resolutions of missing label problems

on the stability of sequential updates and downdates the updating and downdating of cholesky decompositions has important applications in a number of areas there is essentially one standard updating algorithm based on plane rotations which is backward stable three downdating algorithms have been treated in the literature the linpack algorithm the method of hyperbolic transformations and chambers 1971 algorithm although none of these algorithms is backward stable the first and third satisfy a relational stability condition it is shown that relational stability extends to a sequence of updates and downdates in consequence other things being equal if the final decomposition in the sequence is well conditioned it will be accurately computed even though intermediate decompositions may be almost completely inaccurate these results are also applied to the two sided orthogonal decompositions such as the urv decomposition

coherent state theory of the noncompact symplectic group an extended coherent state theory is presented for the noncompact sp 3 r group which reveals a simple relationship between the sp 3 r algebra and its contracted u 3 boson limit the relationship is used to derive a remarkably accurate analytic expression for sp 3 r matrix elements for the generic lowest weight representations the expression is shown to be exact whenever the states involved are multiplicity free with respect to the u 3 subalgebra it is further shown how exact matrix elements are easily calculated in general dyson and holstein primakoff type u 3 boson expansions are given

multipliers and singular integrals on exponential growth groups we propose a simple abstract version of calderon zygmund theory which is applicable to spaces with exponential volume growth and then show that important specific operators can be treated within this framework

an updating algorithm for subspace tracking in certain signal processing applications it is required to compute the null space of a matrix whose rows are samples of a signal with p components the usual tool for doing this is the singular value decomposition however the singular value decomposition has the drawback that it requires o p sup 3 operations to recompute when a new sample arrives it is shown that a different decomposition called the urv decomposition is equally effective in exhibiting the null space and can be updated in o p sup 2 time the updating technique can be run on a linear array of p processors in o p time

spectral multipliers on a class of na groups with rank two we consider a family of real na groups with rank two and we prove that these groups have sub laplacians with differentiable lp functional calculus for all p 1

modified gram schmidt based methods for block downdating the cholesky factorization the hyperbolic modified gram schmidt hmgs method is proposed for block downdating the cholesky factorization the method might be unsatisfactory due to rounding errors a modified version based on the mgs process is presented and is shown to be mixed stable numerical tests show that the new method has the same numerical properties as the generalized linpack type algorithm and can work better than the householder based algorithm given by bojanczyk and steinhardt 1991 9

vector coherent state representation theory a vector coherent state theory is formulated as a natural extension of standard coherent state theory it is shown that the godement representations and the coherent state representations of the sp n r groups of rowe and of deenen and quesne are special cases of this more general theory

vector coherent state theory as a theory of induced representations a general definition is given of vector coherent state vcs representation theory it is shown that the theory is more general than suggested by previous applications and that it incorporates the standard theories of induced representations as special cases the associated k matrix theory is also given a fuller treatment than hitherto and shown to provide a rather general algorithm both for projecting vcs representations from larger representations in which they are embedded and for determining the hermitian form with respect to which an isometric equivalent representation is in fact isometric

matrix elements of operators in symmetrical u 6 u 3 u 2 u 1 and u 6 su 3 so 3 so 2 basis in this paper we consider a basis for n boson states classified by the chain of groups u 6 u 3 u 2 u 1 we determine analytical expressions for the matrix elements of the boson creation and annihilation operators as well as for the u 6 generators with respect to that basis for such purpose we use the wigner eckart theorem with respect to u 3 and calculate the reduced matrix elements of the appropriate irreducible tensors from their matrix elements between highest weight states then we apply the transformation brackets from the chain u 6 u 3 u 2 u 1 to the physical chain u 6 su 3 so 3 so 2 to obtain the matrix elements of the above mentioned operators in a basis corresponding to the latter the matrix elements so determined can be used either in the nuclear interacting boson model of arima and iachello or in the microscopic nuclear collective model of vanagas et al

determination of the sp 2d r generator matrix elements through a boson mapping for the discrete series irreducible representations lambda n 2 d of sp 2d r the determination of the sp 2d r generator matrix elements in an sp 2d r contains implies u d basis is reduced to the much simpler calculation of boson operator matrix elements between u nu contains implies u d boson states where nu d d 1 2 the key of this reduction is the previously derived holstein primakoff boson representation of the sp 2d r generators as an illustration the case of sp 6 r is worked out in detail

vector coherent state theory of the generic representations of so 5 in an so 3 basis for applications of group theory in quantum mechanics one generally needs explicit matrix representations of the spectrum generating algebras that arise in bases that reduce the symmetry group of some hamiltonian of interest here we use vector coherent state techniques to develop an algorithm for constructing the matrices for arbitrary finite dimensional irreps of the so 5 lie algebra in an so 3 basis the so 3 subgroup of so 5 is defined by regarding so 5 as linear transformations of the five dimensional space of an so 3 irrep of angular momentum two a need for such irreps arises in the nuclear collective model of quadrupole vibrations and rotations the algorithm has been implemented in maple and some tables of results are presented

construction of su 3 irreps in canonical so 3 coupled bases alternative methods for defining canonical so 3 coupled bases for su 3 irreps are considered and compared it is shown that a basis that diagonalizes a particular linear combination of so 3 invariants in the su 3 universal enveloping algebra gives basis states that have good k quantum numbers in the asymptotic rotor model limit

canonical orthonormal basis for su 3 so 3 i construction of the basis a canonical orthonormal basis is given for generic representations of the group chain su 3 contains implies so 3

schur type methods for solving least squares problems with toeplitz structure we give an overview of fast algorithms for solving least squares problems with toeplitz structure based on generalization of the classical schur algorithm and discuss their stability properties in order to obtain more accurate triangular factors of a toeplitz matrix as well as accurate solutions for the least squares problems methods based on corrected seminormal equations csne can be used we show that the applicability of the generalized schur algorithm is considerably enhanced when the algorithm is used in conjunction with csne several numerical tests are reported where different variants of the generalized schur algorithm and csne are compared for their accuracy and speed

unifying unitary and hyperbolic transformations abstract in this paper we describe unified formulas for unitary and hyperbolic reflections and rotations and show how these unified transformations can be used to compute a hermitian triangular decomposition r h d r of a strongly nonsingular indefinite matrix a given in the form a x 1 h x 1 x 2 h x 2 1 the unification is achieved by the introduction of signature matrices which determine whether the applicable transformations are unitary hyperbolic or their generalizations we derive formulas for the condition numbers of the unified transformations propose pivoting strategies for lowering the condition number of the transformations and present a unified stability analysis for applying the transformations to a matrix

unitary representations branching rules and matrix elements for the non compact symplectic groups the complementarity of the symplectic and orthogonal groups is used to infer properties of the infinite dimensional unirreps of the former from the character theory of the latter the complete set of d series metaplectic unirreps of sp n r is identified and branching rules are given for their restrictions to the maximal compact subgroup u n developed in terms of the properties of schur functions a known algorithm for the evaluation of matrix elements of the sp 3 r lie algebra is extended to any sp n r and analytic expressions are given for important classes of unirreps and multiplicity free states

vector coherent state representations of so 5 in an so 3 basis vector coherent state theory is used to give an algorithm for constructing the so 5 irreps of type n 0 and n n in the physical so 3 basis the construction is remarkably simple and permits the matrices of the so 5 operators for irreps of low dimension and or small multiplicities to be computed by hand the construction is applied to the 2 0 and 2 2 irreps to illustrate the process

internal labelling operators and contractions of lie algebras we analyse under which conditions the missing label problem associated with a reduction chain of simple lie algebras can be completely solved by means of an inonu wigner contraction naturally related to the embedding this provides a new interpretation of the missing label operators in terms of the casimir operators of the contracted algebra and shows that the available labelling operators are not completely equivalent further the procedure is used to obtain upper bounds for the number of invariants of affine lie algebras arising as contractions of semi simple algebras

racah s method for general subalgebra chains coupling coefficients of so 5 in canonical and physical bases it is shown that the method of infinitesimal generators racah s method can be broadly and systematically formulated as a method applicable to the calculation of reduced coupling coefficients for a generic subalgebra chain g h provided the reduced matrix elements of the generators of g and the recoupling coefficients of h are known the calculation of so 5 so 4 reduced coupling coefficients is considered as an example and a procedure for transformation of reduced coupling coefficients between canonical and physical subalegebra chains is presented the problem of calculating coupling coefficients for generic irreps of so 5 reduced with respect to any of its subalgebra chains is completely resolved by this approach

weak type 1 1 estimates for heat kernel maximal functions on lie groups for a lie group g with left invariant haar measure and associated lebesgue spaces lp g we consider the heat kernels pt t o arising from a right invariant laplacian a on g that is u t pt f solves the heat equation a ot a u 0 with initial condition u o f we establish weak type 1 1 estimates for the maximal operator xf ji f supt 0 pt f 1 and for related hardy littlewood maximal operators in a variety of contexts namely for groups of polynomial growth and for a number of classes of iwasawa an groups we also study the local maximal operator x0 of sup0 o t ipt f i and related hardy littlewood operators for all lie groups

vector coherent state theory in a group with non commuting raising generators vector coherent state theory is used to give an explicit construction of the irreducible representations of a group with non commuting raising generators the simple example of the compact unitary symplectic group usp 4 contains implies usp 2 usp 2

dynamical group of microscopic collective states i one dimensional case in the present series of papers it is intended to determine the nature and study various realizations of the dynamical group of microscopic collective states for an a nucleon system defined as those a particle states invariant under the orthogonal group o n associated with the n a 1 jacobi vectors the present paper discusses the case of a hypothetical one dimensional space simple invariance considerations show that the dynamical group of collective states is then the group spc 2 r which is the restriction to the collective subspace of the group sp 2 r of linear canonical transformations in n dimensions conserving the o n symmetry in addition to the well known realization of the dynamical group in the schrodinger representation based upon the dzublik zickendraht transformation two new realizations are proposed the first acts in a barut hilbert space which is the subspace of a bargmann hilbert space of analytic functions left invariant by o n a unitary mapping is established between the ordi

a rotor expansion of the su 3 lie algebra vector coherent state vcs theory is used to give a rotor expansion of the su 3 lie algebra in a way that parallels the boson expansions that have been made for other lie algebras the construction provides a systematic procedure for calculating hermitian matrix elements of su 3 in an so 3 coupled basis and represents a new development in vcs theory and in the theory of induced representations

harmonic analysis on semigroups wherever t 0 and 1 p so e m stein st 3 developed a littlewoodpaley theory for such semigroups with some additional hypotheses r r coifman r rochberg and g weiss crw pointed out that some of stein s estimates could be obtained more simply by the use of transference techniques which were developed by coifman and weiss cw our object is to push these transference methods to present an alternative and simpler approach to some of the principal results of stein s book we shall also use these results to prove estimates for the wave equation

models for su 3 in terms of so n 2 and so 2n algebras mathematical and physical models for su 3 are given in terms of unirreps of the so n 2 and so 2n algebras vector coherent state representations are given for all the discrete series unirreps of so n 2 and so 2n traceless bosons are shown to arise naturally in these representations a new and elegant realisation of the so 8 approximately so 6 2 model for su 3 is given and it is shown how to perform calculations in the so 8 model using the powerful vector coherent state representations formalism

an estimate for a first order riesz operator on the affine group on the affine group of the line which is a solvable lie group of exponential growth we consider a right invariant laplacian a for a certain right invariant vector field x we prove that the first order riesz operator xa 1 2 is of weak type 1 1 with respect to the left haar measure of the group this operator is therefore also bounded on lp 1 p 2 locally the operator is a standard singular integral the main part of the proof therefore concerns the behaviour of the kernel of the operator at infinity and involves cancellation

haar like expansions and boundedness of a riesz operator on a solvable lie group consider the group of affine transformations xi mapsto e t xi s of the line denote by x and y the right invariant vector fields corresponding to the s and t directions respectively and let delta x 2 y 2 we prove that the first order riesz operator delta 1 2 x is of weak type 1 1 with respect to left haar measure this operator is therefore also bounded on l p 1 p infty our results provide answers in a particular instance to the open question of the boundedness of riesz operators on lie groups of exponential growth the main parts of the proof concern the behaviour of the kernel of the operator at infinity and exploit cancellation a key technique is to use expansion with respect to scales of haar like functions

vector coherent state representations induced representations and geometric quantization i scalar coherent state representations coherent state theory is shown to reproduce three categories of representations of the spectrum generating algebra for an algebraic model i classical realizations which are the starting point for geometric quantization ii induced unitary representations corresponding to prequantization and iii irreducible unitary representations obtained in geometric quantization by choice of a polarization these representations establish an intimate relation between coherent state theory and geometric quantization in the context of induced representations

stable subspace tracking algorithm based on a signed urv decomposition subspace estimation and tracking are of fundamental importance in many signal processing algorithms the class of schur subspace estimators provides a complete parametrization of all principal subspace estimates defined as the column spans of corresponding low rank matrix approximants that lie within a specified 2 norm distance of a given matrix the parametrization is found in terms of a two sided hyperbolic decomposition hyperbolic urv or hurv which can be computed using hyperbolic rotations unfortunately such rotations are commonly associated with numerical instabilities in this paper we present a numerically stable non iterative algorithm to compute the hurv called the signed urv surv algorithm we show that this algorithm implicitly imposes certain constraints on the hurv such that important norm bounds that guarantee stability are satisfied the constraints also restrict the parametrization of the subspace estimate such that it becomes close to the principal subspace provided by the svd which is a special case within this class the complexity of the algorithm is of the same order as that of a qr update updating and downdating are of the same complexity and are both numerically stable surv is proven to provide rank estimates consistent with the svd with the same rank threshold it can replace an svd where only subspace estimation is needed typical applications would e g be the detection of the number of signals in array signal processing and subspace estimation for source separation and interference mitigation such as the first step in music and esprit type algorithms simulation results demonstrate the numerical stability and confirm that this algorithm provides exact rank estimates and good principal subspace estimates as compared to the svd

dyadic sets maximal functions and applications on ax b groups let s be the lie group mathbb r n ltimes mathbb r where mathbb r acts on mathbb r n by dilations endowed with the left invariant riemannian symmetric space structure and the right haar measure which is a lie group of exponential growth hebisch and steger in math z 245 37 61 2003 proved that any integrable function on s admits a calderon zygmund decomposition which involves a particular family of sets called calderon zygmund sets in this paper we show the existence of a dyadic grid in the group s which has nice properties similar to the classical euclidean dyadic cubes using the properties of the dyadic grid we prove a fefferman stein type inequality involving the dyadic hardy littlewood maximal function and the dyadic sharp function as a consequence we obtain a complex interpolation theorem involving the hardy space h1 and the space bmo introduced in collect math 60 277 295 2009

heisenberg weyl algebras of symmetric and antisymmetric bosons all the polynomials in symmetric 2 and antisymmetric 11 u n bosons are constructed in u n bases and u n reduced matrix elements for the bosons between polynomial basis states are computed applications to representation theory of lie groups paired fermion and boson physics are briefly discussed

stability analysis and fast algorithms for triangularization of toeplitz matrices a new o mn algorithm for triangularizing an m times n toeplitz matrix is presented the algorithm is based on the previously developed recursive algorithms that exploit the toeplitz structure and compute each row of the triangular factor via updating and downdating steps a forward error analysis for this existing recursive algorithm is presented which allows us to monitor the conditioning of the problem and use the method of corrected semi normal equations to obtain higher accuracy for certain ill conditioned matrices numerical experiments show that the new algorithm improves the accuracy significantly while the computational complexity stays in o mn

representation of the five dimensional harmonic oscillator with scalar valued u 5 so 5 so 3 coupled vcs wave functions vector coherent state methods which reduce the u 5 so 5 so 3 subgroup chain are used to construct basis states for the five dimensional harmonic oscillator algorithms are given to calculate matrix elements in this basis the essential step is the construction of so 5 so 3 irreps of type v 0 the methodology is similar to that used in two recent papers except that one dimensional as opposed to multidimensional vector valued wave functions are used to give conceptually simpler results another significant advance is a canonical resolution of the so 5 so 3 multiplicity problem

vector coherent state theory and its application to the orthogonal groups vector coherent state theory is developed and presented in a form that explicitly exhibits its general applicability to the ladder representations of all semisimple lie groups and their lie algebras it is shown that in a suitable basis the vector coherent state inner product can be inferred algebraically by k matrix theory and changed to a simpler bargmann inner product thereby facilitating the explicit calculation of the matrix representaions of lie algebras applications are made to the even and odd orthogonal lie algebras

clebsch gordan coefficients of su 3 in su 2 and so 3 bases new algorithms are developed for the purpose of optimizing the efficient calculation of su 3 clebsch gordan coefficients in both su 2 and so 3 coupled bases the new algorithms make use of the fact that highest weight states in a tensor product space are easily identified by vector coherent state methods the methods are developed for su 3 but apply to other compact semi simple lie groups

spectral multipliers on exponential growth solvable lie groups let m be a measure space and let l be a positive definite operator on l2 m by the spectral theorem for any bounded borel measurable function f 0 7 c the operator f l f 0 f de f is bounded on l2 m we are interested in sufficient conditions on f for f l to be bounded on lp m p 6 2 we direct the reader to 1 3 4 8 9 10 12 and 13 for more background on various multiplier theorems in this paper we assume f is compactly supported and have some smoothness finite number of derivatives and we consider only the case p 1 our measure space g is semidirect product of stratified nilpotent lie group n and the real line the operator l is minus sublaplacian on g our group has exponential volume growth the earlier theory suggested that one needs holomorphic f for f l to be bounded on l1 however the recent results 5 6 7 showed that estimates on only a finite number of derivatives of f imply boundedness of f l on l1 on some solvable g of exponential growth in this case we say that g more precisely l has ckfunctional calculus on the other hand christ and muller give an example of a solvable lie group on which f must be holomorphic the problem is to find the condition on g and possibly l which decides whether g has a ck functional calculus or not here our condition is in terms of roots of adjoint representation of the lie algebra of g our groups are of rank one

sub laplacians of holomorphic lp type on rank one an groups and related solvable groups consider a right invariant sub laplacian l on an exponential solvable lie group g endowed with a left invariant haar measure depending on the structure of g and possibly also that of l l may admit differentiable lp functional calculi or may be of holomorphic lp type for a given p 2 as recent studies of specific classes of groups g and sub laplacians l have revealed by holomorphic lp type we mean that every lp spectral multiplier for l is necessarily holomorphic in a complex neighborhood of some point in the l2 spectrum of l this can only arise if the group algebra l1 g is non symmetric in this article we prove that for large classes of exponential groups including all rank one an groups a certain lie algebraic condition which characterizes the non symmetry of l1 g 37 also suffices for l to be of holomorphic l1 type moreover if this condition which was first introduced by j boidol 6 in a different context holds for generic points in the dual g of the lie algebra of g then l is of holomorphic lp type for every p 2 besides the non symmetry of l1 g also the closedness of coadjoint orbits plays a crucial role we also discuss an example of a higher rank an group this example and our results in the rank one case suggest that sub laplacians on exponential lie groups may be of holomorphic l1 type if and only if there exists a closed coadjoint orbit g such that the points of satisfy boidol s condition in the course of the proof of our main results whose principal strategy is similar as in 8 we develop various tools which may be of independent interest and largely apply to more general lie groups some of them are certainly known as folklore results for instance we study subelliptic estimates on representation spaces the relation between spectral multipliers and unitary representations and develop some holomorphic and continuous perturbation theory for images of sub laplacians under smoothly varying families of irreducible unitary representations

analytical expressions for the matrix elements of the non compact symplectic algebra analytic matrix elements are derived for all the lowest weight representations of the noncompact symplectic algebra sp 3 r it is shown that the expressions are exact for representations of the type sigma 1 sigma 2 sigma 3 and for all states of an arbitrary representation that are multiplicity free with respect to the u 3 subalgebra furthermore they are remarkably accurate in general