the euclidean root of snell s law i geometric polarization optics coset spaces of the euclidean group of rigid motions in three space are used as a model of the rays in geometric polarization optics the haar measure invariance leads to canonicity of the phase space transformations the phase space of optical rays undergoes a canonical map also under the effect of smooth refracting or reflecting surfaces between two optical media that is governed by snell s law this is a conservation law between two irreducible representations of the euclidean group labeled by the refractive indices n n of the media these surface transformations are canonical and furthermore factorize into two root transformations that are also canonical this factorization applies equally to the transformation of the polarization vector the root transformation permits the computation of an aberration expansion of the polarization field over the object screen the generators of infinitesimal surface transformations are found i e those that transform between media n and n dn it is shown that

asymptotic type for sectorial operators and an integral of fractional powers abstract there is a standard notion of type for a sectorial linear operator acting in a banach space we introduce a notion of asymptotic type for a linear operator v involving estimates on the resolvent i v 1 as 0 we show for example that if v is sectorial and of asymptotic type then the fractional power v is of asymptotic type for a suitable range of positive moreover we establish various properties of the operator 0 1 d v in particular this operator is of asymptotic type 0 for a sectorial operator v this result has an application to the construction of operators satisfying the well known ritt resolvent condition

ramanujan formula for the generalized stirling approximation abstract the aim of this paper is to improve ramanujan s formula for approximation of the factorial function starting from burnside s formula in contradistinction with the classical formula that starts from stirling s formula

poincare group and relativistic wave equations in dimensions using the generalized regular representation an explicit construction of the unitary irreducible representations of the 2 1 poincare group is presented a detailed description of the angular momentum and spin in 2 1 dimensions is given on this base the relativistic wave equations for all spins including fractional are constructed

functional calculus estimates for tadmor ritt operators abstract we show h functional calculus estimates for tadmor ritt operators also known as ritt operators which generalize and improve results by vitse these estimates are in conformity with the best known power bounds for tadmor ritt operators in terms of the constant dependence furthermore it is shown how discrete square function estimates influence the functional calculus estimates

subordinated discrete semigroups of operators given a power bounded linear operator t in a banach space and a probability f on the non negative integers one can form a subordinated operator s k 0 f k t k we obtain asymptotic properties of the subordinated discrete semigroup s n n 1 2 under certain conditions on f in particular we study probabilities f with the property that s satisfies the ritt resolvent condition whenever t is power bounded examples and counterexamples of this property are discussed the hypothesis of power boundedness of t can sometimes be replaced by the weaker kreiss resolvent condition

a besov class functional calculus for bounded holomorphic semigroups abstract it is well known that 2 sectorial operators generally do not admit a bounded h calculus over the right half plane in contrast to this we prove that the h calculus is bounded over any class of functions whose fourier spectrum is contained in some interval e with 0 e the constant bounding this calculus grows as log e e as e and this growth is sharp over all banach space operators of the class under consideration it follows from these estimates that 2 sectorial operators admit a bounded calculus over the besov algebra b 1 0 of the right half plane we also discuss the link between 2 sectorial operators and bounded tadmor ritt operators

jacobi polynomials bernstein type inequalities and dispersion estimates for the discrete laguerre operator abstract the present paper is about bernstein type estimates for jacobi polynomials and their applications to various branches in mathematics this is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of schrodinger equations whose hamiltonian is given by the generalized laguerre operator more precisely we show that dispersive estimates for the schrodinger equation associated with the generalized laguerre operator are connected with bernstein type inequalities for jacobi polynomials we use known uniform estimates for jacobi polynomials to establish some new dispersive estimates in turn the optimal dispersive decay estimates lead to new bernstein type inequalities

quantum mechanics on p adic numbers a quantum system with positions in and momenta in is studied displacement operators in the phase space of this system and the corresponding heisenberg weyl group are studied it is shown that such a system can be constructed from a semi infinite chain of spins which are coupled in a particular way

meixner pollaczek polynomials and the heisenberg algebra an alternative proof is given for the connection between a system of continuous hahn polynomials and identities for symmetric elements in the heisenberg algebra which was first observed by bender mead and pinsky phys rev lett 56 2445 1986 j math phys 28 509 1987 the continuous hahn polynomials turn out to be meixner pollaczek polynomials use is made of the connection between laguerre polynomials and meixner pollaczek polynomials the rodrigues formula for laguerre polynomials an operational formula involving meixner pollaczek polynomials and the schrodinger model for the irreducible unitary representations of the three dimensional heisenberg group

on sic povms in prime dimensions the generalized pauli group and its normalizer the clifford group have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete povms sic povms to date almost every known sic povm fiducial vector is an eigenstate of a canonical unitary in the clifford group i show that every canonical unitary in prime dimensions p 3 lies in the same conjugacy class of the clifford group and give a class representative for all such dimensions it follows that if even one such sic povm fiducial vector is an eigenvector of such a unitary then all of them are for a given such dimension i also conjecture that in all dimensions d the number of conjugacy classes is bounded above by 3 and depends only on dmod9 and i support this claim with computer computations in all dimensions 48

the group theoretical treatment of aberrating systems i aligned lens systems in third aberration order the constituents of a lens system i e slabs of homogeneous transparent material and the refracting surfaces between them considered to third aberration order are associated to elements of a nine parameter aberration group three parameters correspond to gaussian systems and six to group classified aberrations the group multiplication operation through matrix cum vector algebra corresponds to their concatenation and the linear group action on an eight dimensional homogeneous space corresponds to the nonlinear action of the system on the optical phase space this leads to economical computation algorithms that may be extended to aberrating systems of higher order

coherent states and the role of the affine group in the quantum mechanics of the morse potential the coherent states of the morse potential that have been obtained earlier from supersymmetric quantum mechanics are shown to be connected with the representations of the affine group of the real line and some of its extensions this relation is similar to the one between the heisenberg weyl group and the coherent states of the harmonic oscillator the states that minimize the uncertainty product of the generators of the affine lie algebra are shown to contain all the coherent states of the morse oscillator plus the intelligent states of the morse hamiltonians with different shape parameter s the representations of the central extension of the affine group denoted by g0 and its further extension will be shown to define the phase space relevant to the problem by choosing an appropriate orbit of the coadjoint representation of this allows one to construct a generalized wigner function on this phase space which is again essentially in the same relation with the affine group as the ordinary wigner function with the heisenberg weyl group

holomorphic semi groups and the geometry of banach spaces according to a fundamental theorem of dvoretzky cf 24 25 any infinite dimensional banach space x contains for each integer n and each e 0 a subspace xn which is 1 e isomorphic to 12n in the classical banach spaces lp and 1p with 1 p x it is well known that this result can be improved these spaces contain uniformly complemented ln 7 s this means that we can find subspaces xn as above with the additional property that there exist projections p x xn such that supn 11 pn 11 cx moreover it is also well known that this property no longer holds for 11 or lo this motivated the question cf 13 to determine which infinite dimensional banach spaces contain uniformly complemented 1 n s we will prove below that all uniformly convex spaces have this property moreover we will show that a banach space x verifies this property locally see definition 2 10 for more precision if and only if x does not contain l n s uniformly progressively the work of previous authors cf 14 12 and particularly 5 has reduced such so called geometric questions to a purely analytic problem consisting of verifying in x a certain inequality referred to as k convexity in this paper we connect this inequality with the holomorphy of a certain semi group of operators on lp x and we show how this leads to the results already mentioned

the group theoretical treatment of aberrating systems iii the classification of asymmetric aberrations a lie theoretical classification of the aberrations of systems modeled by asymmetric optical devices is given the classification is done on the basis of aberration order and axial symmetry of the first order part this leads to finite dimensional nonunitary representations of sp 4 r reduced with respect to its sp 2 r subalgebra with helicity and symplectic spin labels based on pure magnifier systems a weight label reproduces and completes seidel s traditional classification of axis symmetric aberrations based on other first order systems such as optical fibers other classification schemes are indicated

an algebraic and geometric approach to non bijective quadratic transformations this work is concerned with an extension of the so called levi civita and kustaanheimo stiefel transformations the extension is achieved along two lines firstly the authors examine the latter transformations in other dimensions than the ones originally considered by on the one hand levi civita and on the other hand by kustaanheimo and stiefel secondly they pass from the compact to the non compact case this leads to quadratic non bijective transformations that they refer to as hurwitz or kustaanheimo stiefel like and quasihurwitz or levi civita like transformations the hurwitz and quasihurwitz transformations are introduced and studied in an algebraic framework which relies on the use of eight dimensional cayley dickson algebras an explicit formulation of the hurwitz and quasihurwitz transformations is also given in terms of clifford algebras the hurwitz transformations are investigated from a geometrical viewpoint indeed they are connected to hopf and pseudohopf fibrations finally some differential aspects of the hurwitz and to a lesser extent quasihurwitz transformations are developed in view of future physical applications

some results on the parametrization of complex hadamard matrices in this paper we provide an analytical procedure which leads to a system of n 2 2 polynomial equations whose solutions give the parametrization of the complex n n hadamard matrices it is shown that in general the hadamard matrices depend on a number of arbitrary phases and a lower bound for this number is given the moduli equations define interesting geometrical objects whose study will shed light on the parametrization of the hadamard matrices as well as on some interesting geometrical varieties defined by them

the pauli matrices in n dimensions and finest gradings of simple lie algebras of type a n 1 properties of the lie algebra gl n c are described for a basis which is a generalization of the 2 2 pauli matrices the 3 3 case is described in detail the remarkable properties of that basis are the grading of the lie algebra it offers each grading subspace is one dimensional and the matrix group it generates it is a finite group with the center of sl n c as its commutator group

mutually unbiased bases and hadamard matrices of order six we report on a search for mutually unbiased bases mubs in six dimensions we find only triplets of mubs and thus do not come close to the theoretical upper bound 7 however we point out that the natural habitat for sets of mubs is the set of all complex hadamard matrices of the given order and we introduce a natural notion of distance between bases in hilbert space this allows us to draw a detailed map of where in the landscape the mub triplets are situated we use available tools such as the theory of the discrete fourier transform to organize our results finally we present some evidence for the conjecture that there exists a four dimensional family of complex hadamard matrices of order 6 if this conjecture is true the landscape in which one may search for mubs is much larger than previously thought

on the ramanujan lodge harmonic number expansion the aim of this paper is to extend and refine the ramanujan lodge harmonic number expansion into negative powers of a triangular number we construct a faster asymptotic series and some new sharp inequalities for the harmonic numbers

functional calculus under the tadmor ritt condition and free interpolation by polynomials of a given degree for banach space operators t satisfying the tadmor ritt condition zi t 1 c z 1 1 z 1 we prove that the best possible constant ct n bounding the polynomial calculus for t p t ct n p deg p n behaves in the worst case as logn as n this result is based on a new free carleson type interpolation theorem for polynomials of a given degree

variations on a theme of heisenberg pauli and weyl the parentage between weyl pairs the generalized pauli group and the unitary group is investigated in detail we start from an abstract definition of the heisenberg weyl group on the field and then switch to the discrete heisenberg weyl group or generalized pauli group on a finite ring the main characteristics of the latter group an abstract group of order d3 noted pd are given conjugacy classes and irreducible representation classes or equivalently lie algebra of dimension d3 associated with pd leaving the abstract sector a set of weyl pairs in dimension d is derived from a polar decomposition of su 2 closely connected to angular momentum theory then a realization of the generalized pauli group pd and the construction of generalized pauli matrices in dimension d are revisited in terms of weyl pairs finally the lie algebra of the unitary group u d is obtained as a subalgebra of the lie algebra associated with pd this leads to a development of the lie algebra of u d in a basis consisting of d2 generalized pauli matrices in the case where d is a power of a prime integer the lie algebra of su d can be decomposed into d 1 cartan subalgebras

exactly solvable position dependent mass hamiltonians related to non compact semi simple lie groups we suggest a generalized procedure to obtain exactly solvable position dependent mass hamiltonians in one dimension the second order casimir invariant of the regular representation of a non compact semi simple lie group g the spectral properties of which are well known is used to introduce exactly solvable hamiltonians a brief description of the procedure is presented and its application to quantum systems associated with sl 2 r is detailed

discrete repulsive oscillator wavefunctions for the study of infinite discrete systems on phase space the three dimensional lorentz algebra and group so 2 1 and so 2 1 provide a discrete model of the repulsive oscillator its eigenfunctions are found in the principal irreducible representation series where the compact generator that we identify with the position operator has the infinite discrete spectrum of the integers z while the spectrum of energies is a double continuum the right and left moving wavefunctions are given by hypergeometric functions that form a dirac basis for 2 z under contraction the discrete system limits to the well known quantum repulsive oscillator numerical computations of finite approximations raise further questions on the use of dirac bases for infinite discrete systems

the unitary irreducible representations of sl 2 r in all subgroup reductions we use the canonical transform realization of sl 2 r in order to find all matrix elements and integral kernels for the unitary irreducible representations of this group explicit results are given for all mixed bases and subgroup reductions these provide the full multiparameter set of integral transforms and series expansions associated to sl 2 r

the metaplectic group within the heisenberg weyl ring the heisenberg weyl ring contains the metaplectic group of canonical transforms acting unitarily on l 2 r these ring elements are characterized through i the integral transform kernels ii coset distributions and iii classical functions under any quantization scheme the isomorphism under group composition leads to several new relations involving twisted products and quantization of gaussian classical functions the wigner inversion operator is a special central group element it is shown that the only quantization scheme invariant under metaplectic transformations is the weyl scheme the structure studied here appears to be relevant to the study of wave optics with aberration

wigner functions and separability for finite systems a discussion of discrete wigner functions in phase space related to mutually unbiased bases is presented this approach requires mathematical assumptions which limits it to systems with density matrices defined on complex hilbert spaces of dimension pn where p is a prime number with this limitation it is possible to define a phase space and wigner functions in close analogy to the continuous case that is we use a phase space that is a direct sum of n two dimensional vector spaces each containing p2 points this is in contrast to the more usual choice of a two dimensional phase space containing p2n points a useful aspect of this approach is that we can relate complete separability of density matrices and their wigner functions in a natural way we discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd special attention is required for two qubits p 2 and our technique fails to establish the separability property for more than two qubits finally we give a brief discussion of hamiltonian dynamics in the language developed in the paper

the heisenberg weyl group in the coherent state basis and the bargmann transform it is shown that the representation matrices of the quantum mechanical group in the coherent state basis can be used as bases for the expansion of arbitrary square integrable functions on the group although this basis is nonorthogonal the expansion can be inverted in a manner similar to an orthogonal basis next a coherent state wave function is defined in analogy with the schrodinger wave function and the above matrix elements are used to determine the action of the group in the space of the coherent state wave functions the function space in this realization is bargmann s hilbert space of analytic functions it is shown that the mixed basis matrix element between the coordinate and coherent state bases is essentially the integral kernel of bargmann the transpose of this matrix element yields the kernel for the inversion of the transform in this construction the bargmann transform appears as the unitary transformation connecting the schrodinger and coherent state wave functions this considerably

an angular momentum approach to quadratic fourier transform hadamard matrices gauss sums mutually unbiased bases the unitary group and the pauli group the construction of unitary operator bases in a finite dimensional hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of su 2 a single formula for the bases is obtained from a polar decomposition of su 2 and is analyzed in terms of cyclic groups quadratic fourier transforms hadamard matrices and generalized gauss sums weyl pairs generalized pauli operators and their application to the unitary group and the pauli group naturally arise in this approach

a generalization of sectorial and quasi sectorial operators abstract in the paper we generalize the main results presented in bentkus and paulauskas 2004 2 by giving rates of approximation of some semigroups of operators of the order n 0 1 also two classes of operators generalizing sectorial and quasi sectorial operators are introduced and their properties are studied

lie algebras for systems with mixed spectra i the scattering poschl teller potential starting from an n body quantum space we consider the lie algebraic framework where the poschl teller hamiltonian 1 2 2 c sech2 s csch2 is the single sp 2 r casimir operator the spectrum of this system is mixed it contains a finite number of negative energy bound states and a positive energy continuum of free states it is identified with the clebsch gordan series of the d d representation coupling the wave functions are the sp 2 r clebsch gordan coefficients of that coupling in the parabolic basis using only lie algebraic techniques we find the asymptotic behavior of these wave functions for the special pure trough potential s 0 we derive thus the transmission and reflection amplitudes of the scattering matrix

linear transformations and aberrations in continuous and finite systems in geometric optics there is a natural distinction between the paraxial and aberration regimes which contain respectively the linear and nonlinear canonical transformations of position and momentum in the phase space in the lie theoretical presentation linear inhomogeneous transformations are generated by linear and quadratic functions of the phase space while aberrations of increasing order are generated by homogeneous functions with higher powers of these coordinates in a way parallel but distinct from the schrodinger quantization of continuous classical systems we quantize the geometric optical model into discrete finite dimensional systems based on the lie algebra whose wavefunctions are n point signals phase space is a sphere and transformations are represented by the n n unitary matrices that form the group we factor this group into linear and nonlinear unitary transformations of phase space and classify all its n2 4 aberrations this offers a new parametrization of based on a chosen subgroup

laplace and segal bargmann transforms on hermitian symmetric spaces and orthogonal polynomials let d g k be a complex bounded symmetric domain of tube type in a complex jordan algebra v and let dr j d d be its real form in a formally real euclidean jordan algebra j v dr h l is a bounded realization of the symmetric cone in j we consider representations of h that are gotten by the generalized segal bargmann transform from a unitary g space of holomorphic functions on d to an l2 space on dr we prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the laplace transform we find the extension to d of the spherical functions on dr and find their expansion in terms of the l spherical polynomials on d which are jack symmetric polynomials we prove that the coefficients are orthogonal polynomials in an l2 space the measure being the harish chandra plancherel measure multiplied by the symbol of the berezin transform we prove the difference equation and recurrence relation for those polynomials by considering the action of the lie algebra and the cayley transform on the polynomials on d finally we use the laplace transform to study generalized laguerre functions on symmetric cones

on measuring unboundedness of the h calculus for generators of analytic semigroups we investigate the boundedness of the h infty calculus by estimating the bound b e of the mapping h infty b x f f a t e for e near zero here a generates the analytic semigroup t and h infty is the space of bounded analytic functions on a domain strictly containing the spectrum of a we show that b e o loge in general whereas b e o 1 for bounded calculi this generalizes a result by vitse and complements work by haase and rozendaal for non analytic semigroups we discuss the sharpness of our bounds and show that single square function estimates yield b e o loge

about the sharpness of the stability estimates in the kreiss matrix theorem one of the conditions in the kreiss matrix theorem involves the resolvent of the matrices a under consideration this so called resolvent condition is known to imply for all n 1 the upper bounds an ek n 1 and an ek n 1 here is the spectral norm k is the constant occurring in the resolvent condition and the order of a is equal to n 1 1 it is a long standing problem whether these upper bounds can be sharpened for all fixed k 1 to bounds in which the right hand members grow much slower than linearly with n 1 and with n 1 respectively in this paper it is shown that such a sharpening is impossible the following result is proved for each e 0 there are fixed values c 0 k 1 and a sequence of n 1 n 1 matrices an satisfyirig the resolvent condition such that an n c n 1 1 e c n 1 1 e for n n 1 2 3 the result proved in this paper is also relevant to matrices a whose e pseudospectra lie at a distance not exceeding ke from the unit disk for all e 0

extremal growth of powers of operators satisfying resolvent conditions of kreiss ritt type let e be a compact subset of the unit circle we determine the extremal rate of growth of tn n 1 for banach space operators t satisfying the resolvent condition t i 1 const dist e 1 this includes as extreme cases the kreiss condition e t and the ritt condition e 1 for intermediate sets e the cardinality the measure and the hausdorff dimension of e all play a role in determining the growth of tn as a by product we also obtain lower bounds for the taylor coefficients of functions f holomorphic on the unit disk and satisfying f z 1dist z e z 1

remarks on a dynamical higher order theory of laminated plates and its application in random vibration response abstract this paper presents an analysis of the equations governing the dynamics of sheardeformable composite plates without recourse to a variational procedure it is noted that the operator associated with the governing equations is nonsymmetric using a first order perturbation technique it is shown to be positive definite in addition using the biorthogonality condition the dynamic response of the plate is formulated

waveform mapping and time frequency processing of dna and protein sequences current state of the art approaches for biological sequence querying and alignment require preprocessing and lack robustness to repetitions in the sequence in addition these approaches do not provide much support for efficiently querying subsequences a process that is essential for tracking localized database matches we propose a query based alignment method for biological sequences that first maps sequences to time domain waveforms before processing the waveforms for alignment in the time frequency plane the mapping uses waveforms such as gaussian functions with unique sequence representations in the time frequency plane the proposed alignment method employs a robust querying algorithm that utilizes a time frequency signal expansion whose basis function is matched to the basic waveform in the mapped sequences the resulting wavequery approach was demonstrated for both deoxyribonucleic acid dna and protein sequences using the matching pursuit decomposition as the signal basis expansion we specifically evaluated the alignment localization of wavequery over repetitive database segments and we demonstrated its operation in real time without preprocessing we also demonstrated that wavequery significantly outperformed the biological sequence alignment method blast for queries with repetitive segments for dna sequences a generalized version of the wavequery approach with the metaplectic transform is also described for protein sequence structure prediction

orthogonal stochastic duality functions from lie algebra representations we obtain stochastic duality functions for specific markov processes using representation theory of lie algebras the duality functions come from the kernel of a unitary intertwiner between representations which provides generalized orthogonality relations for the duality functions in particular we consider representations of the heisenberg algebra and mathfrak su 1 1 both cases lead to orthogonal self duality functions in terms of hypergeometric functions for specific interacting particle processes and interacting diffusion processes

on time dependent quadratic quantum hamiltonians we apply the techniques of canonical transforms to equations of the type left a t mathbb p 2 b t mathbb p mathbb q mathbb q mathbb p c t mathbb q 2 d t mathbb q e t mathbb p f t mathbb 1 right psi q t i partial t psi q t where mathbb q and mathbb p are the quantum position and momentum operators the time dependent parameters of the text w wedge text sl text 2 r evolution operator are found through linear differential equations in terms of these we give explicitly the green s function all separating coordinates and similarity solutions of the equation we analyze the behavior of gaussian and coherent state initial conditions in closed form and present a new interpretation of all the lewis riesenfeld constants of motion

the clebsch gordan coefficients of the three dimensional lorentz algebra in the parabolic basis starting from the oscillator representation of the three dimensional lorentz algebra so 2 1 we build a lie algebra of second order differential operators which realizes all series of self adjoint irreducible representations the choice of the common self adjoint extention over a two chart function space determines whether they lead to single or multivalued representations over the corresponding lie group the diagonal operator defining the basis is the parabolic subgroup generator the direct product of two such algebras allows for the calculation of all clebsch gordan coefficients explicitly as solutions of schrodinger equations for poschl teller potentials over one d d two d c or three c c charts all coefficients are given in terms of up to two 2f1 hypergeometric functions

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

group theoretical construction of mutually unbiased bases in hilbert spaces of prime dimensions mutually unbiased bases in hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity an alternative proof of existence of a maximal collection of n 1 mutually unbiased bases in hilbert spaces of prime dimension n is given by exploiting the finite heisenberg group also called the pauli group and the action of on finite phase space implemented by unitary operators in the hilbert space crucial for the proof is that for prime is also a finite field

the isotropic lines of bb z _ d 2 we show that the isotropic lines in the lattice are the lagrangian submodules of that lattice and we give their number together with the number of them through a given point of the lattice the set of isotropic lines decomposes into orbits under the action of we give an explicit description of those orbits as well as their number and respective cardinalities we also develop two group actions on the group related to the topic

the gel fand realization and the exceptional representations of sl 2 r it is shown that the canonical representation space of gel fand and co workers is particularly appropriate for problems requiring explicit reduction under the noncompact so 1 1 and e 1 bases for both the principal and exceptional series of representations of sl 2 r we use this realization to set up complete orthonormal sets of eigendistributions corresponding to the three subgroup reductions namely sl 2 r so 1 1 sl 2 r e 1 and sl 2 r so 2 and evaluate the unitary transformations connecting these reductions these overlap matrix elements appear as the applications of these distributions to a set of well defined test functions using the rigorous theory of analytic continuation we show that the results for the exceptional representations have the same analytic forms as the corresponding results for the principal series some of these results are essential prerequisites for the solution of the clebsch gordan problem series and coefficients of sl 2 r in the so 1 1 basis

symmetric informationally complete positive operator valued measures and the extended clifford group we describe the structure of the extended clifford group defined to be the group consisting of all operators unitary and antiunitary which normalize the generalized pauli group or weyl heisenberg group as it is often called we also obtain a number of results concerning the structure of the clifford group proper i e the group consisting just of the unitary operators which normalize the generalized pauli group we then investigate the action of the extended clifford group operators on symmetric informationally complete positive operator valued measures or sic povms covariant relative to the action of the generalized pauli group we show that each of the fiducial vectors which has been constructed so far including all the vectors constructed numerically by renes et al is an eigenvector of one of a special class of order 3 clifford unitaries this suggests a strengthening of a conjecture of zauner s we give a complete characterization of the orbits and stability groups in dimensions 2 7 fina