cca secure encryption supporting authorized equality test on ciphertexts in standard model and its applications abstract we present an encryption scheme for authorized equality test on ciphertexts seet which allows the data owner to authorize a tester to compare her ciphertexts without decrypting their values the security of seet is formally proved against three types of adversary two of them for ciphertext confidentiality in the phases before and after authorization respectively and the third for token privacy to the best of our knowledge our seet construction is the first encryption scheme supporting equality test on ciphertexts that is proven secure against the three types of adversary in the standard model our seet construction outperforms existing schemes in terms of ciphertext size and encryption decryption testing costs to show its application in set operations we extend it into schemes for controlled set distance computation such that a curious server is able to deduce the similarity dissimilarity score between two encrypted user sets without knowing their elements

moyal like form of the star product for generalized su 2 stratonovich weyl symbols the star product is the basic tool used in the phase space formulation of quantum mechanics we find a differential form of the star product for a class of mbox s parametrized su 2 stratonovich weyl symbols which appear in the phase space representation of spin like systems the limit of large spin is considered and the asymptotic form of the differential operator defining the star product is obtained

commuting periodic operators and the periodic wigner function commuting periodic operators cpo depending on the coordinate and the momentum operators are defined the cpo are functions of the two basic commuting operators and with a being an arbitrary constant a periodic wigner function pwf w x p is defined and it is shown that it is applicable in a normal expectation value calculation to the cpo as done in the original wigner paper moreover this pwf is non negative everywhere and it can therefore be interpreted as an actual probability distribution the pwf w x p is shown to be given as an expectation value of the periodic dirac delta function in the phase plane

wigner weyl correspondence in quantum mechanics for continuous and discrete systems a dirac inspired view drawing inspiration from dirac s work on functions of non commuting observables we develop an approach to phase space descriptions of operators and the wigner weyl correspondence in quantum mechanics complementary to standard formulations this involves a two step process introducing phase space descriptions based on placing position dependences to the left of momentum dependences or the other way around then carrying out a natural transformation to eliminate a kernel which appears in the expression for the trace of the product of two operators the method works uniformly for both continuous cartesian degrees of freedom and for systems with finite dimensional state spaces it is interesting that the kernel encountered is naturally expressible in terms of geometric phases and its removal involves extracting its square root in a suitable manner

digital computation of linear canonical transforms we deal with the problem of efficient and accurate digital computation of the samples of the linear canonical transform lct of a function from the samples of the original function two approaches are presented and compared the first is based on decomposition of the lct into chirp multiplication fourier transformation and scaling operations the second is based on decomposition of the lct into a fractional fourier transform followed by scaling and chirp multiplication both algorithms take n log n time where n is the time bandwidth product of the signals the only essential deviation from exactness arises from the approximation of a continuous fourier transform with the discrete fourier transform thus the algorithms compute lcts with a performance similar to that of the fast fourier transform algorithm in computing the fourier transform both in terms of speed and accuracy

identity based encryption with outsourced equality test in cloud computing it is the first time to integrate identity based encryption into public key encryption with equality test it extends identity based encryption with keyword search to yield a general function equality test it is proven to be one way chosen ciphertext security against a chosen identity attack as a desirable security unlike peks or ibeks it can be used alone since it contains decryption algorithm besides encryption algorithm we firstly combine the concepts of public key encryption with equality test pkeet and identity based encryption ibe to obtain identity based encryption with equality test ibeet inheriting the advantage of ibe ibeet can simplify the certificate management of pkeet with all messages encrypted with the receiver s public identity in the ibeet scheme the receiver computes a trapdoor using the secret value for the identity and then sends it to a cloud server for equality test on its ciphertexts with others ciphertexts using this primitive someone with the trapdoor for its identity can delegate out the capability of equality test on its ciphertexts without requiring a central authority to act as a delegator so it is very suitable for the client with minimal computation resource e g mobile phone furthermore compared with pkeet it has security improvement since not anyone can perform the test therefore ibeet may have interesting applications in cloud computing e g partition of encrypted emails we define one way chosen ciphertext security against a chosen identity attack ow id cca and propose a construction in bilinear pairing finally extensive security analysis and comparison with related works show that the proposed scheme is proven secure and useful

finite two dimensional oscillator ii the radial model a finite two dimensional radial oscillator of n 1 2 points is proposed with the dynamical lie algebra so 4 su 2 x su 2 y examined in part i of this work but reduced by a subalgebra chain so 4 so 3 so 2 as before there are a finite number of energies and angular momenta the casimir spectrum of the new chain provides the integer radii 0 n and the 2 1 discrete angles on each circle are obtained from the finite fourier transform of angular momenta the wavefunctions of the finite radial oscillator are so 3 clebsch gordan coefficients we define here the hankel hahn transforms with dual hahn polynomials as finite n unitary approximations to hankel integral transforms with bessel functions obtained in the contraction limit n

quantum algebraic structures compatible with the harmonic oscillator newton equation we study some of the algebraic structures that are compatible with the quantization of the harmonic oscillator through its newton equation examples of such structures are given they include undeformed and q deformed oscillators as well as the su 2 and the deformed suq 2 lie algebras which appear in a variety of physical models

wigner functions for curved spaces ii on spheres the form of the wigner distribution function for hamiltonian systems in spaces of constant negative curvature i e hyperboloids proposed in m a alonso g s pogosyan and k b wolf wigner functions for curved spaces i on hyperboloids j math phys 43 5857 2002 is extended here to spaces whose curvature is constant and positive i e spheres an essential part of this construction is the use of the functions of sherman and volobuyev which are an overcomplete set of plane wave like solutions of the laplace beltrami equation for this space rotations that displace the poles transform these functions with a multiplier factor and their momentum direction becomes formally complex the covariance properties of the proposed wigner function are understood in these terms as an example for the one dimensional case we consider the energy eigenstates of the oscillator on the circle in a poschl teller potential the standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature

invariant inner products on spaces of solutions of the klein gordon and helmholtz equations we construct sesquilinear forms which are invariant under the similarity groups of the klein gordon and helmholtz equations these give rise to positive definite inner products on subspaces of solutions of these equations for the klein gordon case the known poincare invariant inner product for positive energy solutions is recovered for the helmholtz case a new euclidean invariant inner product is presented which involves the function and its normal derivative to a line integrated nonlocally over that line

prepotential formulation of su 3 lattice gauge theory the su 3 lattice gauge theory is reformulated in terms of su 3 prepotential harmonic oscillators this reformulation has enlarged su 3 u 1 u 1 gauge invariance under which the prepotential operators transform like matter fields the hilbert space of su 3 lattice gauge theory is shown to be equivalent to the hilbert space of the prepotential formulation satisfying certain color invariant sp 2 r constraints the su 3 irreducible prepotential operators which solve these sp 2 r constraints are used to construct su 3 gauge invariant hilbert spaces at every lattice site in terms of su 3 gauge invariant vertex operators the electric fields and the link operators are reconstructed in terms of these su 3 irreducible prepotential operators we show that all the su 3 mandelstam constraints become local and take a very simple form within this approach we also discuss the construction of all possible linearly independent su 3 loop states which solve the mandelstam constraints the techniques can be easily generalized to su n

the schwinger su 3 construction i multiplicity problem and relation to induced representations the schwinger oscillator operator representation of su 3 is analyzed with particular reference to the problem of multiplicity of irreducible representations it is shown that with the use of an sp 2 r unitary representation commuting with the su 3 representation the infinity of occurrences of each su 3 irreducible representation can be handled in complete detail a natural generating representation for su 3 containing each irreducible representation exactly once is identified within a subspace of the schwinger construction and this is shown to be equivalent to an induced representation of su 3

continuous vs discrete fractional fourier transforms abstract we compare the finite fourier exponential and fourier kravchuk transforms both are discrete finite versions of the fourier integral transform the latter is a canonical transform whose fractionalization is well defined we examine the harmonic oscillator wavefunctions and their finite counterparts mehta s basis functions and the kravchuk functions the fractionalized fourier kravchuk transform was proposed in j opt soc amer a 14 1997 1467 1477 and is here subject of numerical analysis in particular we follow the harmonic motions of coherent states within a finite discrete optical model of a shallow multimodal waveguide

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

wigner distributions for finite state systems without redundant phase point operators we set up wigner distributions for n state quantum systems following a dirac inspired approach in contrast to much of the work in this study requiring a 2n 2n phase space particularly when n is even our approach is uniformly based on an n n phase space grid and thereby avoids the necessity of having to invoke a quadrupled phase space and hence the attendant redundance both n odd and even cases are analysed in detail and it is found that there are striking differences between the two while the n odd case permits full implementation of the marginal property the even case does so only in a restricted sense this has the consequence that in the even case one is led to several equally good definitions of the wigner distributions as opposed to the odd case where the choice turns out to be unique

su n coherent states and irreducible schwinger bosons we exploit the su n irreducible schwinger boson to construct su n coherent states this construction of an su n coherent state is analogous to the construction of the simplest heisenberg weyl coherent states the coherent states belonging to irreducible representations of su n are labeled by the eigenvalues of n 1 su n casimir operators and characterized by n 1 complex orthonormal vectors describing the su n group manifold

the wigner function of the relativistic finite difference oscillator in an external field the phase space representation for a relativistic linear oscillator in a homogeneous external field expressed through the finite difference equation is constructed explicit expressions of the relativistic oscillator wigner quasi distribution function for the stationary states as well as for the states of thermodynamical equilibrium are obtained and their correct limits are shown

jacobi ϑ functions and discrete fourier transforms properties of the jacobi 3 function and its derivatives under discrete fourier transforms are investigated and several interesting results are obtained the role of modulo n equivalence classes in the theory of functions is stressed an important conjecture is studied

morse theory by perturbation methods with applications to harmonic maps there are many interesting variational problems for which the palaissmale condition cannot be verified in cases where the palais smale condition can be verified for an approximating integral and the critical points converge a morse theory is valid this theory applies to a class of variational problems consisting of the energy integral for harmonic maps with a lower order potential the abstract morse theory of palais and smale 9 can be applied with success to many problems in the calculus of variations to get existence theorems for stationary or critical points of integrals from topological information however there is a natural limitation in these applications in that the conditions are stated in terms of a fixed function space which will usually be a sobolev space the integral must both be differentiable and satisfy a norm convergence property which palais and smale call condition c in the same space of functions in this article we introduce a perturbation method which circumvents this difficulty in some variational problems this technique may be applied in particular to harmonic mappings and we include this somewhat technical discussion in the last section the author is grateful to g k francis and r s palais for a number of helpful conversations 1 palais smale morse theory in this article b will denote a banach manifold with at least a c2 structure modeled on a banach space with at least c2 partitions of unity we assume that this manifold is equipped with a c1 finsler structure in which the manifold is complete if a function f is defined and differentiable on b ldfxi maxovet b idf v i iv2 is well defined we say that the function f satisfies palais smale condition c if for any set s c b on which if s i is bounded and idf s i is not bounded away from zero the closure s contains a critical point of f a point x where dfx 0 we have the following theorem due to palais and schwartz 10 12 1 1 theorem let b be a complete c2 finsler manifold without boundary andf b r a c2 function satisfying palais smale condition c then f satisfies a lusternik schnirelman theory in particular a if f is bounded below it takes on its minimum on every component of b and there are at least as many critical points as cat b received by the editors march 13 1979 and in revised form september 26 1979 1980 mathematics subject classification primary 49f15 secondary 58e05 research supported in part by the national science foundation 1981 american mathematical society 0002 9947 81 0000 0462 04 75 569 this content downloaded from 157 55 39 159 on sun 18 sep 2016 06 10 27 utc all use subject to http about jstor org terms

wigner transform and pseudodifferential operators on symmetric spaces of non compact type we obtain a general expression for a wigner transform wigner function on symmetric spaces of non compact type and study the weyl calculus of pseudodifferential operators on them

wigner distributions and quantum mechanics on lie groups the case of the regular representation we consider the problem of setting up the wigner distribution for states of a quantum system whose configuration space is a lie group the basic properties of wigner distributions in the familiar cartesian case are systematically generalized to accommodate new features which arise when the configuration space changes from n dimensional euclidean space rn to a lie group g the notion of canonical momentum is carefully analyzed and the meanings of marginal probability distributions and their recovery from the wigner distribution are clarified for the case of compact g an explicit definition of the wigner distribution is proposed possessing all the required properties geodesic curves in g which help introduce a notion of the midpoint of two group elements play a central role in the construction

wigner distribution function for euclidean systems euclidean systems include poly and monochromatic wide angle optics acoustics and also infinite discrete data sets we use a recently defined wigner operator and quasiprobability distribution function to set up and study the phase space evolution of these models subject to differential and difference equations respectively infinite data sets and two dimensional monochromatic helmholtz fields are thus shown by their wigner function on a cylinder of direction and location the wigner function for polychromatic wavefields has c number coordinates of two dimensional wavenumber and position

quantizations on the circle and coherent states we present a possible construction of coherent states on the unit circle as configuration space our approach is based on borel quantizations on s1 including the aharonov bohm type quantum description coherent states are constructed by perelomov s method as group related coherent states generated by weyl operators on the quantum phase space because of the duality of canonical coordinates and momenta i e the angular variable and the integers this formulation can also be interpreted as coherent states over an infinite periodic chain for the construction we use the analogy with our quantization and coherent states over a finite periodic chain where the quantum phase space was the coherent states constructed in this work are shown to satisfy the resolution of unity to compare them with canonical coherent states some of their further properties are also studied demonstrating similarities as well as substantial differences this article is part of a special issue of journal of physics a mathematical and theoretical devoted to coherent states mathematical and physical aspects

a generalized wigner function for quantum systems with the su 2 dynamical symmetry group we introduce a wigner like quasidistribution function to describe quantum systems with the su 2 dynamic symmetry group this function is defined in a three dimensional group manifold and can be used to represent the states defined in several su 2 invariant subspaces the explicit differential moyal like form of the star product is found and analyzed in the semiclassical limit

on the number of minimal two spheres of small area in manifolds with curvature bounded above a beautiful application of the morse theory of closed geodesics is provided by klingenberg s k 1 study of the number of short smooth closed geodesics in sufficiently pinched spheres together with subsequent contributions in this direction by many authors a particularly striking result see ballmann thorbergsson and ziller btz states that a generic metric on a compact simply connected riemannian manifold whose sectional curvatures satisfy 1 4 k a 1

a generalized wigner function on the space of irreducible representations of the weyl heisenberg group and its transformation properties a natural extension of the wigner function to the space of irreducible unitary representations of the weyl heisenberg group is discussed the action of the automorphisms group of the weyl heisenberg group onto wigner functions and their generalizations and onto symplectic tomograms is elucidated some examples of physical systems are considered to illustrate some aspects of the characterization of the wigner functions as solutions of differential equations

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

continuum quantum systems as limits of discrete quantum systems i state vectors abstract dynamical systems on continuum hilbert spaces may be realized as limits of dynamical systems on discrete possibly finite dimensional hilbert spaces in this first of four papers on the topic the continuum and discrete spaces are interfaced to one another algebraically convergence of vectors is defined in such a way as to preserve inner products and a necessary and sufficient coordinate wise criterion for convergence is proved

finite two dimensional oscillator i the cartesian model a finite two dimensional oscillator is built as the direct product of two finite one dimensional oscillators using the dynamical lie algebra su 2 x su 2 y the position space in this model is a square grid of points while the ordinary continuous two dimensional quantum oscillator has a symmetry algebra u 2 the symmetry algebra of the finite model is only u 1 x u 1 y because it lacks rotations in the position and momentum plane we show how to import an so 2 group of rotations from the continuum model that transforms unitarily the finite wavefunctions on the fixed square grid we thus propose a finite analogue for fractional u 2 fourier transforms

general approach to mathfrak su n quasi distribution functions we propose an operational form for the kernel of a mapping between an operator acting in a hilbert space of a quantum system with an symmetry group and its symbol in the corresponding classical phase space for symmetric irreps of this mapping is bijective we briefly discuss complications that will occur in the general case

wigner functions for curved spaces i on hyperboloids we propose a wigner quasiprobability distribution function for hamiltonian systems in spaces of constant curvature in this article on hyperboloids which returns the correct marginals and has the covariance of the shapiro functions under so d 1 transformations to the free systems obeying the laplace beltrami equation on the hyperboloid we add a conic oscillator potential in the hyperbolic coordinate as an example we analyze the one dimensional case on a hyperbola branch where this conic oscillator is the poschl teller potential we present the analytical solutions and plot the computed results the standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature

continuum quantum systems as limits of discrete quantum systems iv affine canonical transforms affine canonical transforms complex order fourier transforms and their associated coherent states appear in two scenarios finite discrete and continuum we examine the relationship between the two scenarios making systematic use of inductive limits which were developed in the preceding articles in this series

existence of closed geodesics on positively curved manifolds on applique la theorie de lusternik schnirelmann pour obtenir l existence de plusieurs geodesiques fermees entre certains niveaux de longueur on etudie les proprietes de stabilite de ces geodesiques fermees

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

minimal two spheres and the topology of manifolds with positive curvature on totally isotropic two planes there has been much interest among differential geometers in finding relationships between curvature and topology of riemannian manifolds for the most part efforts have been directed towards understanding the implications of positive or negative sectional curvature ricci curvature or scalar curvature but there are other hypotheses on the curvature which also deserve investigation in this article we will consider the topological implications of a new curvature assumption positive curvature on totally isotropic two planes and we will prove via the sacks uhlenbeck theory of minimal two spheres that a compact simply connected riemannian manifold of dimension at least four with positive curvature on totally isotropic two planes is homeomorphic to a sphere as corollaries we will obtain a proof via minimal two spheres for riemannian manifolds of dimension at least four of the berger klingenberg rauch toponogov sphere theorem strengthened to pointwise quarter pinching as well as a proof that every simply connected compact riemannian manifold with positive curvature operators is homeomorphic to a sphere let m be an n dimensional riemannian manifold with tangent space tpm at the point p e m recall that the curvature operator at p is the self adjoint

tomograms and other transforms a unified view a general framework is presented which unifies the treatment of wavelet like quasidistribution and tomographic transforms explicit formulae relating the three types of transforms are obtained the case of transforms associated with the symplectic and affine groups is treated in some detail special emphasis is given to the properties of the scale time and scale frequency tomograms tomograms are interpreted as a tool to sample the signal space by a family of curves or as the matrix element of a projector

su n irreducible schwinger bosons we construct su n irreducible schwinger bosons satisfying certain u n 1 constraints which implement the symmetries of su n young tableaues as a result all su n irreducible representations are simple monomials of n 1 types of su n irreducible schwinger bosons further we show that these representations are free of multiplicity problems thus all su n representations are made as simple as su 2

wigner distribution function for finite systems we construct a wigner distribution function for finite data sets it is based on a finite optical system a linear wave guide where the finite number of discrete sensors is equal to the number of modes which the guide can carry the dynamical group for this model is su 2 and the wave functions are sets of n 2l 1 data points the wigner distribution function assigns classical c numbers to the operators of position momentum and wave guide mode

position and momentum bases for the monochromatic maxwell fish eye and the sphere in geometric optics the maxwell fish eye is a medium where light rays follow circles while in scalar wave optics this medium can only trap fields of certain discrete frequencies in the monochromatic case characterized by a positive integer l there are independent fields we identify two bases of functions one known as the sherman volobuyev functions is characterized as of most definite momenta the other is new and composed of most definite positions and normal derivatives for the fish eye scalar wavefields their construction uses the stereographic projection of the sphere and their identification is corroborated in the contraction limit to the homogeneous helmholtz medium

semi generic construction of public key encryption and identity based encryption with equality test public key encryption with equality test pkeet which was first introduced by yang et al ct rsa 2010 has various applications including facilitating keyword search on encrypted data and partitioning encrypted data on the cloud it can be also applied to manage personal health records on the internet for these reasons there have been improvements on earlier pkeet schemes in terms of performance and functionality we present a semi generic method for pkeet constructions assuming only the existence of ind cca2 secure traditional public key encryption pke schemes the hardness of computational diffie hellman cdh problems and random oracles our approach has several advantages it enables us to understand requirements for the equality test functionality more clearly furthermore our approach is quite general in that if we change the underlying pke scheme with the identity based encryption ibe scheme and we assume the hardness of bilinear diffie hellman problems instead of cdh then we obtain the first ibe scheme with equality test ibeet satisfying analogous security arguments to those of pkeet although an ibeet construction was recently proposed but we note that it satisfies only weak security requirements

the schwinger su 3 construction ii relations between heisenberg weyl and su 3 coherent states the schwinger oscillator operator representation of su 3 studied in a previous paper from the representation theory point of view is analyzed to discuss the intimate relationships between standard oscillator coherent state systems and systems of su 3 coherent states both su 3 standard coherent states based on choice of highest weight vector as fiducial vector and certain other specific systems of generalized coherent states are found to be relevant a complete analysis is presented covering all the oscillator coherent states without exception and amounting to su 3 harmonic analysis of these states

coherent states for su 3 we define coherent states for su 3 using six bosonic creation and annihilation operators these coherent states are explicitly characterized by six complex numbers with constraints for the completely symmetric representations n 0 and 0 m only three of the bosonic operators are required for mixed representations n m all six operators are required the coherent states provide a resolution of identity satisfy the continuity property and possess a variety of group theoretic properties we introduce an explicit parametrization of the group su 3 and the corresponding integration measure finally we discuss the path integral formalism for a problem in which the hamiltonian is a function of su 3 operators at each site

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

continuum quantum systems as limits of discrete quantum systems ii state functions in this second of four papers on the eponymous topic pointwise convergence of a discrete state function to a continuum state function is shown to imply the algebraic criterion for convergence that was introduced in the prequel as examples and as a prerequisite for the sequels the normal approximation theorem and the convergence of the kravchuk functions to the hermite gaussians are expressed in terms of the algebraic notion of convergence

bumpy metrics and closed parametrized minimal surfaces in riemannian manifolds the purpose of this article is to study conformal harmonic maps f m where is a closed riemann surface and m is a compact riemannian manifold of dimension at least four such maps define parametrized minimal surfaces possibly with branch points we show that when the ambient manifold m is given a generic metric all prime closed parametrized minimal surfaces are free of branch points and are as morse nondegenerate as allowed by the group of automorphisms of e they are morse nondegenerate in the usual sense if has genus at least two lie on two dimensional nondegenerate critical submanifolds if s has genus one and on six dimensional nondegenerate critical submanifolds if has genus zero

anonymous hibe compact construction over prime order groups we present an anonymous hierarchical identity based encryption hibe scheme that not only achieves constant size ciphertext but also works in bilinear groups with prime order in contrast to our work all previous anonymous hibe schemes either have ciphertext size linear in the depth of the hierarchy or work in bilinear groups with a composite order our construction is more efficient than previous schemes in terms of ciphertext size and decryption cost we give provable security of our construction consisting of confidentiality and anonymity without random oracles to gain anonymity we introduce a novel and simple method called double exponent technique which gives an anonymous hibe scheme more compact than the result at crypto 2006 our method is quite general in that it works on both symmetric and asymmetric bilinear maps the simple method might also be of independent interest to gain anonymity of attributes in constructing searchable encryption schemes on encrypted data

continuum quantum systems as limits of discrete quantum systems iii operators convergence of a discrete operator to a continuum operator is defined as examples the circular rotor the one dimensional box the harmonic oscillator and the fractional fourier transform are realized as limits of finite dimensional quantum systems limits thus defined preserve algebraic structure the results prepare for a sequel in which some affine canonical transforms will be discretized

minimal two spheres of low index in manifolds with positive complex sectional curvature suppose that s n is given a generic riemannian metric with sectional curvatures which satisfy a suitable pinching condition formulated in terms of complex sectional curvatures this pinching condition is satisfied by manifolds whose real sectional curvatures k r sigma satisfy begin aligned 1 2 k r sigma le 1 end aligned then the number of minimal two spheres of morse index lambda for n 2 le lambda le 2n 5 is at least p 3 lambda n 2 where p 3 k is the number of k cells in the schubert cell decomposition for g 3 mathbb r n 1

spectra of phase point operators in odd prime dimensions and the extended clifford group we analyze the role of the extended clifford group in classifying the spectra of phase point operators within the framework laid out by gibbons et al phys rev a 70 062101 2004 for setting up wigner distributions on discrete phase spaces based on finite fields to do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field auxiliary results include a derivation of the conjugacy classes of esl 2 fn

wigner weyl isomorphism for quantum mechanics on lie groups the wigner weyl isomorphism for quantum mechanics on a compact simple lie group g is developed in detail several features are shown to arise which have no counterparts in the familiar cartesian case notable among these is the notion of a semiquantized phase space a structure on which the weyl symbols of operators turn out to be naturally defined and figuratively speaking located midway between the classical phase space t g and the hilbert space of square integrable functions on g general expressions for the star product for weyl symbols are presented and explicitly worked out for the angle angular momentum case