explicit representations of changeable degree spline basis functions changeable degree spline cd spline for short basis functions defined by an iterative integral method are extensions of b spline basis functions a cd spline basis function is a piecewise function made up of polynomials of different degrees in this paper we will give the explicit representations of cd spline basis functions from which the spanned linear space can be seen clearly our method is also feasible for the explicit representations of the other basis functions given in analogous integral ways

weighted complex projective 2 designs from bases optimal state determination by orthogonal measurements we introduce the problem of constructing weighted complex projective 2 designs from the union of a family of orthonormal bases if the weight remains constant across elements of the same basis then such designs can be interpreted as generalizations of complete sets of mutually unbiased bases being equivalent whenever the design is composed of d 1 bases in dimension d we show that for the purpose of quantum state determination these designs specify an optimal collection of orthogonal measurements using highly nonlinear functions on abelian groups we construct explicit examples from d 2 orthonormal bases whenever d 1 is a prime power covering dimensions d 6 10 and 12 for example where no complete sets of mutually unbiased bases have thus far been found

nearly defective complex eigenvalues in bessel matrices abstract it is known that an n n bessel matrix t n exhibits simple well separated complex eigenvalues which are getting more and more ill conditioned for increasing n the aim of this paper is to study the behavior of such eigenvalues which are simple in theory but in practice become defective using finite precision to this end we provide a sharp estimate of the distance of t n to the nearest defective matrix when n increases

symmetric informationally complete positive operator valued measures a new computer study we report on a new computer study of the existence of d2 equiangular lines in d complex dimensions such maximal complex projective codes are conjectured to exist in all finite dimensions and are the underlying mathematical objects defining symmetric informationally complete measurements in quantum theory we provide numerical solutions in all dimensions d 67 and moreover a putatively complete list of weyl heisenberg covariant solutions for d 50 a symmetry analysis of this list leads to new algebraic solutions in dimensions d 24 35 and 48 which are given together with algebraic solutions for d 4 15 and 19

multicomplementary operators via finite fourier transform a complete set of d 1 mutually unbiased bases exists in a hilbert space of dimension d whenever d is a power of a prime we discuss a simple construction of d 1 disjoint classes each one having d 1 commuting operators such that the corresponding eigenstates form sets of unbiased bases such a construction works properly for prime dimension we investigate an alternative construction in which the real numbers that label the classes are replaced by a finite field having d elements one of these classes is diagonal and can be mapped to cyclic operators by means of the finite fourier transform which allows one to understand complementarity in a similar way as for the position momentum pair in standard quantum mechanics the relevant examples of two and three qubits and two qutrits are discussed in detail

characterization and construction of the nearest defective matrix via coalescence of pseudospectral components let a be a matrix with distinct eigenvalues and let w a be the distance from a to the set of defective matrices using either the 2norm or the frobenius norm define the pseudospectrum of a to be the set of points in the complex plane which are eigenvalues of matrices a e with kek and let c a be the supremum of all with the property that has n distinct components demmel and wilkinson independently observed in the 1980s that w a c a and equality was established for the 2 norm by alam and bora in 2005 we give new results on the geometry of the pseudospectrum near points

galois quantum systems irreducible polynomials and riemann surfaces finite quantum systems in which the position and momentum take values in the galois field gf pl are studied ideas from the subject of field extension are transferred in the context of quantum mechanics the frobenius automorphisms in galois fields lead naturally to the frobenius formalism in a quantum context the hilbert space splits into frobenius subspaces which are labeled with the irreducible polynomials associated with the ypl y the frobenius maps transform unitarily the states of a galois quantum system and leave fixed all states in some of its galois subsystems where the position and momentum take values in subfields of gf pl an analytic representation of these systems in the l sheeted complex plane shows deeper links between galois theory and riemann surfaces

matrix polynomials with specified eigenvalues abstract this work concerns the distance in the 2 norm from a given matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane initially we consider perturbations of the constant coefficient matrix only a singular value optimization characterization is derived for the associated distance we also consider the distance in the general setting when all of the coefficient matrices are perturbed in this general setting we obtain a lower bound in terms of another singular value optimization problem the singular value optimization problems derived facilitate the numerical computation of the distances

on condition numbers of polynomial eigenvalue problems in this paper we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients generalizing classical results of matrix perturbation theory we provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate we obtain that if a simple eigenvalue of a matrix polynomial is ill conditioned in some respects then it is close to be multiple and we construct an upper bound for this distance measured in the euclidean norm we also derive a new expression for the condition number of a simple eigenvalue which does not involve eigenvectors moreover an elsner like perturbation bound for matrix polynomials is presented

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

on the distance from a matrix polynomial to matrix polynomials with some prescribed eigenvalues abstract consider an n n matrix polynomial p and a set consisting of k n complex numbers recently kokabifar loghmani psarrakos and karbassi studied a weighted spectral norm distance from p to the n n matrix polynomials whose spectra contain the specified set under the assumption that all the entries of are distinct in this paper the case in which some or all of the desired eigenvalues can be multiple is discussed lower and upper bounds for the distance are computed and a perturbation of p associated to the upper bound is constructed a detailed numerical example illustrates the efficiency and validity of the proposed computational method

pseudospectra critical points and multiple eigenvalues of matrix polynomials abstract we develop a general framework for perturbation analysis of matrix polynomials more specifically we show that the normed linear space l m c n n of n by n matrix polynomials of degree at most m provides a natural framework for perturbation analysis of matrix polynomials in l m c n n we present a family of natural norms on the space l m c n n and show that the norms on the spaces c m 1 and c n n play a crucial role in the perturbation analysis of matrix polynomials we define pseudospectra of matrix polynomials in the general framework of the normed space l m c n n and show that the pseudospectra of matrix polynomials well known in the literature follow as special cases we analyze various properties of pseudospectra in the unified framework of the normed space l m c n n we analyze critical points of backward errors of approximate eigenvalues of matrix polynomials and show that each critical point is a multiple eigenvalue of an appropriately perturbed polynomial we show that common boundary points of components of pseudospectra of matrix polynomials are critical points as a consequence we show that a solution of wilkinson s problem for matrix polynomials can be read off from the pseudospectra of matrix polynomials

finite sequences of orthogonal polynomials connected by a jacobi matrix abstract if p0 pn is an orthogonal sequence with pj a monic polynomial of exact degree j all j then the sequence q0 qn of monic polynomials defined by the requirement that pj cjqn j 1pn 1 at the zeros of pn with cj suitable constants is also orthogonal with respect to a possibly differently weighted scalar product these two sequences can be shown to be the upper left respectively lower right principal minors of the matrix t j with j a suitable tridiagonal matrix the sequence q occurs in the characterization of the discrete least squares approximation to f from span p0 pj in terms of the n j 1 order divided differences of f p n 1 at the nodes i e the zeros of pn a complete characterization of the pair u v of zero sets of pj 1 and qn j is given and an application is made to the problem of recovering a jacobi matrix from such data

a formula for the 2 norm distance from a matrix to the set of matrices with multiple eigenvalues we prove that the 2 norm distance from an n times n matrix a to the matrices that have a multiple eigenvalue lambda is equal to where the singular values sigma k are ordered nonincreasingly therefore the 2 norm distance from a to the set of matrices with multiple eigenvalues is

on the construction of nearest defective matrices to a normal matrix given an n by n normal matrix a having n distinct eigenvalues we describe a simple procedure for constructing a defective matrix nearest to a our construction requires only an appropriate pair of eigenvalues of a and their corresponding eigenvectors

symmetric informationally complete quantum measurements we consider the existence in arbitrary finite dimensions d of a positive operator valued measure povm comprised of d2 rank one operators all of whose operator inner products are equal such a set is called a symmetric informationally complete povm sic povm and is equivalent to a set of d2 equiangular lines in cd sic povms are relevant for quantum state tomography quantum cryptography and foundational issues in quantum mechanics we construct sic povms in dimensions two three and four we further conjecture that a particular kind of group covariant sic povm exists in arbitrary dimensions providing numerical results up to dimension 45 to bolster this claim

there is no generalization of known formulas for mutually unbiased bases in a quantum system having a finite number n of orthogonal states two orthonormal bases ai and bj are called mutually unbiased if all inner products ai bj have the same modulus 1 n this concept appears in several quantum information problems the number of pairwise mutually unbiased bases is at most n 1 and various constructions of such n 1 bases have been found when n is a power of a prime number we study families of formulas that generalize these constructions to arbitrary dimensions using finite rings we then prove that there exists a set of n 1 mutually unbiased bases described by such formulas if and only if n is a power of a prime number

locating a nearest matrix with an eigenvalue of prespecified algebraic multiplicity the wilkinson distance of a matrix a is the two norm of the smallest perturbation e so that a e has a multiple eigenvalue malyshev derived a singular value optimization characterization for the wilkinson distance in this work we generalize the definition of the wilkinson distance as the two norm of the smallest perturbation so that the perturbed matrix has an eigenvalue of prespecified algebraic multiplicity we provide a singular value characterization for this generalized wilkinson distance then we outline a numerical technique to solve the derived singular value optimization problems in particular the numerical technique is applicable to malyshev s formula to compute the wilkinson distance as well as to retrieve a nearest matrix with a multiple eigenvalue

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

tomographically complete sets of orthonormal bases in finite systems quantum systems where the position and momentum are in the ring d is an odd integer are considered a tomographically complete set of bases i e a set of bases such that probabilities from tomography experiments corresponding to these bases can be used for the calculation of an arbitrary density matrix is considered such a set of bases is constructed by considering a factorization of a system with dimension d p1p2 where p1 p2 are prime numbers in terms of two subsystems with dimensions p1 and p2 appropriate combination of the mutually unbiased bases in the subsystems leads to a tomographically complete set of bases in the full system this set leads to probabilities along all the lines through the origin 0 0 in the phase space with exactly d points it is shown that the number of these lines is d the dedekind function the general theory is exemplified with examples

discrete wigner function for finite dimensional systems a phase space approach to finite dimensional systems is developed from basic principles for a system describable by a hilbert space of dimension d we define a one to one correspondence between operators and functions on a discrete and finite phase space with points valid for any dimension d the properties fulfilled by this correspondence and its uniqueness are examined this formalism is applied to the number difference and phase difference of a two mode field this case is compared with the marginal distribution for these variables arising from a two mode wigner function for number and phase

the distance from a matrix polynomial to matrix polynomials with a prescribed multiple eigenvalue abstract for a matrix polynomial p and a given complex number we introduce a spectral norm distance from p to the matrix polynomials that have as an eigenvalue of geometric multiplicity at least and a distance from p to the matrix polynomials that have as a multiple eigenvalue then we compute the first distance and obtain bounds for the second one constructing associated perturbations of p

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

conjugate variables in finite phase plane we construct two pairs of quasicoordinates and quasimomenta in a finite phase plane which form sets of conjugate variables in such a plane the coordinate x is quantized with a step c and the momentum p with a step where mc is the size of the phase plane in the x direction the construction depends crucially on the possibility of writing m m1m2 with m1 and m2 relatively prime the conjugate variables are applied to harper like hamiltonians it is shown how to design physical systems with energy spectra containing any desired number of discrete energy levels say m1 each of them having a prescribed degeneracy m2

feynman s path integral and mutually unbiased bases our previous work on quantum mechanics in hilbert spaces of finite dimension n is applied to elucidate the deep meaning of feynman s path integral pointed out by g svetlichny he speculated that the secret of the feynman path integral may lie in the property of mutual unbiasedness of temporally proximal bases we confirm the corresponding property of the short time propagator by using a specially devised n n approximation of quantum mechanics in applied to our finite dimensional analogue of a free quantum particle

factorizations and physical representations a hilbert space in m dimensions is shown explicitly to accommodate representations that reflect the decomposition of m into prime numbers representations that exhibit the factorization of m into two relatively prime numbers the kq representation zak j 1970 phys today 23 51 and related representations termed q1q2 representations together with their conjugates are analysed as well as a representation that exhibits the complete factorization of m in this latter representation each quantum number varies in a subspace that is associated with one of the prime numbers that make up m

quantum systems with finite hilbert space galois fields in quantum mechanics a galois quantum system in which the position and momentum take values in the galois field gf pl is considered it is comprised of l component systems which are coupled in a particular way and is described by a certain class of hamiltonians displacements in the gf pl gf pl phase space and the corresponding heisenberg weyl group are studied symplectic transformations are shown to form the sp 2 gf pl group wigner and weyl functions are defined and their properties are studied frobenius symmetries which are based on frobenius automorphisms in the theory of galois fields are a unique feature of these systems for l 2 if they commute with the hamiltonian there are constants of motion which are discussed an analytic representation in the l sheeted complex plane provides an elegant formalism that embodies the properties of frobenius transformations the difference between a galois quantum system and other finite quantum systems where the position and momentum take values in the ring is discussed

an elsner like perturbation theorem for generalized eigenvalues in 1985 elsner established a general bound on the distance between an eigenvalue of a matrix and the closest eigenvalue of a perturbation of that matrix in this note we show that a similar result holds for the generalized eigenvalue problem

curve construction based on four αβ bernstein like basis functions a b s t r a c t four new bernstein like basis functions with two exponential shape parameters are constructed in this paper which include the cubic said ball basis functions and the cubic bernstein basis functions within the general framework of quasi extended chebyshev space we prove that the proposed bernstein like basis is an optimal normalized totally positive basis in order to compute the corresponding bezier like curves stably and efficiently a new corner cutting algorithm is developed necessary and sufficient conditions are derived for the planar bezier like curve having single or double inflection points a loop or a cusp or be locally or globally convex in terms of the relative position of its control polygons side vectors based on the new proposed bernstein like basis a class of b spline like basis functions with two local exponential shape parameters is constructed their totally positive property is also proved the associated b splinelike curves have c 2 continuity at single knots and include the cubic non uniform b spline

weak mutually unbiased bases quantum systems with variables in are considered the properties of lines in the phase space of these systems are studied weak mutually unbiased bases in these systems are defined as bases for which the overlap of any two vectors in two different bases is equal to d 1 2 or alternatively to one of the d 1 2i 0 where di is a divisor of d apart from d 1 they are designed for the geometry of the phase space in the sense that there is a duality between the weak mutually unbiased bases and the maximal lines through the origin in the special case of prime d there are no divisors of d apart from 1 d and the weak mutually unbiased bases are mutually unbiased bases

multi degree smooth polar splines a framework for geometric modeling and isogeometric analysis we develop a multi degree polar spline framework with applications to both geometric modeling and isogeometric analysis first multi degree splines are introduced as piecewise non uniform rational b splines nurbs of non uniform or variable polynomial degree and a simple algorithm for their construction is presented then an extension to two dimensional polar configurations is provided by means of a tensor product construction with a collapsed edge suitable combinations of these basis functions encoded in a so called isogeometric analysis suitable extraction operator yield ckck smooth polar splines for any k 0k 0 we show that it is always possible to construct a set of smooth polar spline basis functions that form a convex partition of unity and possess locality explicit constructions for k 0 1 2 k 0 1 2 are presented optimal approximation behavior is observed numerically and examples of applications to free form design smooth hole filling and high order partial differential equations demonstrate the applicability of the developed framework

nearest matrix with prescribed eigenvalues and its applications consider an n n matrix a and a set consisting of k n prescribed complex numbers lippert 2010 in a challenging article studied geometrically the spectral norm distance from a to the set of matrices whose spectra included specified set and constructed a perturbation matrix with minimum spectral norm such that a had in its spectrum this paper presents an easy practical computational method for constructing the optimal perturbation by improving and extending the methodology necessary definitions and lemmas of previous related works also some conceivable applications of this issue are provided

an optimal bound for the spectral variation of two matrices abstract we establish a bound for the spectral variation of two complex n n matrices a b in terms of a b and a b here denotes the spectral norm it is always better than a bound previously given by bhatia and friedland and it is optimal we describe the set of pairs a b for which the bound is attained

root neighborhoods of a polynomial the root neighborhoods of p z a polynomial over the complex field are the sets of complex numbers that are the roots of polynomials which are near to p z the term near means that the coefficients of the polynomials are within some fixed magnitude of the coefficients of p z a necessary and sufficient condition for a complex number to be in the root neighborhoods is given and it is proved that each root neighborhood contains at least one root of p z and the same number of roots of each near polynomial finally a necessary and sufficient condition is given for a root neighborhood to contain more than one root of p z and consequently more than one root of any of the near polynomials

the angle angular momentum quantum phase space the angle angular momentum quantum phase space is considered and the corresponding heisenberg weyl group is studied when 2j 1 is a power of a prime z 2j 1 is a galois field and stronger results can be proved sl 2 z 2j 1 transformations are explicitly constructed and various properties of the displacement operators are studied central extensions of the abelian group by z 2j 1 are studied and they provide all the ways of constructing the heisenberg weyl group from g and z 2j 1

on neighbouring matrices with quadratic elementary divisors this paper presents new results on matrices with quadratic elementary divisors neighbouring a given matrixa and relates them to those given by kakan it shows that all results of this kind may give severe overestimates of the distance to the nearest defective matrix and that matrices with elementary divisors of degrees higher than two are often at a much greater distance than those with quadratic divisors

factorization in finite quantum systems unitary transformations in an angular momentum hilbert space h 2j 1 are considered they are expressed as a finite sum of the displacement operators which play the role of su 2j 1 generators with the weyl function as coefficients the chinese remainder theorem is used to factorize large qudits in the hilbert space h 2j 1 in terms of smaller qudits in hilbert spaces h 2ji 1 all unitary transformations on large qudits can be performed through appropriate unitary transformations on the smaller qudits

a general class of bernstein like bases this paper presents a unified approach to deal with spaces containing simultaneously algebraic and trigonometric or hyperbolic polynomials bases with optimal shape preserving and stability properties are constructed evaluation and subdivision algorithms for them are provided bases for the corresponding and mixed spline spaces are also constructed some nice properties of these bases and the generated curves are shown

hjelmslev geometry of mutually unbiased bases the basic combinatorial properties of a complete set of mutually unbiased bases mubs of a q dimensional hilbert space with p being a prime and r a positive integer are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective hjelmslev plane defined over a galois ring of characteristic p2 and rank r the q vectors of a basis of correspond to the q points of a so called neighbour class and the q 1 mubs answer to the total number of pairwise disjoint neighbour classes on the conic

mutually unbiased bases generalized spin matrices and separability abstract a collection of orthonormal bases for a complex d dimensional hilbert space is called mutually unbiased mub if for any two vectors v and w from different bases the square of the inner product equals 1 d v w 2 1 d the mub problem is to prove or disprove the existence of a maximal set of d 1 bases it has been shown in ann phys 191 1989 363 that such a collection exists if d is a power of a prime number p we revisit this problem and use d d generalizations of the pauli spin matrices to give a constructive proof of this result specifically we give explicit representations of commuting families of unitary matrices whose eigenvectors solve the mub problem additionally we give formulas from which the orthogonal bases can be readily computed we show how the techniques developed here provide a natural way to analyze the separability of the bases the techniques used require properties of algebraic field extensions and the relevant part of that theory is included in the appendix

on pseudospectra of matrix polynomials and their boundaries in the first part of this paper sections 2 4 the main concern is with the boundary of the pseudospectrum of a matrix polynomial and particularly with smoothness properties of the boundary in the second part sections 5 6 results are obtained concerning the number of connected components of pseudospectra as well as results concerning matrix polynomials with multiple eigenvalues or the proximity to such polynomials

projective ring line of an arbitrary single qudit as a continuation of our previous work havlicek and saniga 2007 j phys a math gen 40 f943 52 preprint 0708 4333 an algebraic geometrical study of a single d dimensional qudit is made with d being any positive integer the study is based on an intricate relation between the symplectic module of the generalized pauli group of the qudit and the fine structure of the projective line over the modular ring explicit formulae are given for both the number of generalized pauli operators commuting with a given one and the number of points of the projective line containing the corresponding vector of we find remarkably that a perp set is not a set theoretic union of the corresponding points of the associated projective line unless d is a product of distinct primes the operators are also seen to be structured into disjoint layers according to the degree of their representing vectors a brief comparison with some multiple qudit cases is made

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

geometrical approach to the discrete wigner function in prime power dimensions we analyse the wigner function in prime power dimensions constructed on the basis of the discrete rotation and displacement operators labelled with elements of the underlying finite field we separately discuss the case of odd and even characteristics and analyse the algebraic origin of the non uniqueness of the representation of the wigner function explicit expressions for the wigner kernel are given in both cases

on sensitivity of eigenvalues and eigendecompositions of matrices abstract the main purpose of this paper is to provide solutions of two problems on sensitivity of eigenvalues and eigendecompositions of matrices the first problem is due to wilkinson and it concerns finding the distance from an n n matrix with n distinct eigenvalues to the set of matrices having multiple eigenvalues we also describe how to construct a nearest matrix having a multiple eigenvalue the second problem concerns providing a characterization of the stability of eigendecompositions of matrices and is due to demmel

changeable degree spline basis functions a b spline basis function is a piecewise function of polynomials of equal degree on its support interval this paper extends b spline basis functions to changeable degree spline cd spline for short basis functions each of which may consist of polynomials of different degrees on its support interval the cd spline basis functions possess many b spline like properties and include the b spline basis functions as subcases their corresponding parametric curves called cd spline curves are like b spline curves and also have many good properties if we use the cd spline basis functions to design a curve made up of polynomial segments of different degrees the number of control points may be decreased

hamiltonians of quantum systems with positions and momenta in gf pl a quantum system with positions and momenta in gf pl is considered such a system can be constructed from l smaller systems in which the positions and momenta take values in zp if the hamiltonian of this l partite system is compatible with gf pl the concept of compatibility of a hamiltonian with gf pl allows the quantum formalism in the l partite system to be expressed in terms of galois arithmetic transformations of the basis in gf pl produce unitary transformations of the quantum states which form a representation of gl l zp they are used to define which subset of the general set of hamiltonians in the l partite system is compatible with gf pl

degree elevation of changeable degree spline changeable degree spline cd spline a direct extension of the b spline allows different segments to possess different degrees in this paper we present a method for degree elevation of a cd spline the method is flexible as it applies to any segment of any cd spline elevated by any degree a cd spline can be transformed into a b spline and vice versa in various orders by using this method moreover this degree elevation method is presented as a corner cutting process

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

geometrical approach to mutually unbiased bases we propose a unifying phase space approach to the construction of mutually unbiased bases for a two qubit system it is based on an explicit classification of the geometrical structures compatible with the notion of unbiasedness these consist of bundles of discrete curves intersecting only at the origin and satisfying certain additional properties we also consider the feasible transformations between different kinds of curves and show that they correspond to local rotations around the bloch sphere principal axes we suggest how to generalize the method to systems in dimensions that are powers of a prime