a new view of d 7 clifford algebra a new view of d 7 clifford algebra is described in which starting from a septet of mutually anticommuting clifford operators one introduces a septet of mutually commuting involutions in so 8 these involutions can be seen to arise naturally from a study of ternary vector cross products in eight dimensions

calculation of pseudospectra by the arnoldi iteration the arnoldi iteration usually viewed as a method for calculating eigenvalues can also be used to estimate pseudospectra this possibility may be of practical importance because in applications involving highly nonnormal matrices or operators such as hydrodynamic stability pseudospectra may be physically more significant than spectra

the spectral geometry of the hopf fibration we study the spectral geometry of the hopf fibration and determine the right invariant metrics on for which there exist eigenforms of the laplacian on which pull back to eigenforms of the laplacian on we show that the pull back of the volume form on can be an eigenform of the laplacian on with non zero eigenvalue we show that if is a principal bundle with a bundle metric and that if then eigenvalues cannot change thus eigenvalues do not change for the fibrations and if we also study the corresponding questions in the complex category for the fibration of the hopf manifold

lie algebras under constraints and non bijective canonical transformations the concept of a lie algebra under constraints is developed in connection with the theory of non bijective canonical transformations a finite dimensional vector space m carrying a faithful linear representation of a lie algebra l is mapped into a lower dimensional space m in such a manner that a subalgebra l0 of l is mapped into d l0 0 the lie algebra l under the constraint d l0 0 is the largest subalgebra l1 of l that can be represented faithfully on m if l0 is semisimple then l1 is shown to be the centraliser cent ll0 if l is semisimple and l0 is a one dimensional diagonal subalgebra of a cartan subalgebra of l then l1 is shown to be the factor algebra cent ll0 l0 the later two results are applied to non bijective canonical transformations generalising the kustaanheimo stiefel transformation

from godel quaternions to nonlinear sigma models from the ring of godel quaternions the authors construct new quaternionic groups and a pseudo hopf fibration these can be used to study sigma models valued in a four dimensional hyperboloid particular solutions of these models are also given

on reduction of two degrees of freedom hamiltonian systems by an s1 action and so0 1 2 as a dynamical group reduction by an s1 action is a method of finding periodic solutions in hamiltonian systems which is known rather as the method of averaging such periodic solutions can be reconstructed as s1 orbits by pulling back the critical points of an associated reduced hamiltonian on a reduced phase space along the reduction for hamiltonian systems of two degrees of freedom a geometric setting of the reduction is already accomplished in the case where the reduced phase space is a two sphere in the euclidean space r3 and the reduced hamilton s equations of motion are euler s equations this article deals with the case where the reduced phase space will be a two hyperboloid in the three minkowski space and the reduced hamilton s equations of motion will be euler s equations with respect to the lorentz metric this reduction is associated with su 1 1 symplectic action on the phase space r4 as a consequence of this association the reduced hamiltonian system proves to admit a dynamical group so0 1 2 a w

accuracy assessment for eigencomputations variety of backward errors and pseudospectra abstract we show that backward errors and pseudospectra combined together are useful tools to assess the validity of a computed eigenvalue 1 given a set of admissible perturbations a on a matrix a and a norm on relative or absolute the backward error z for z as a candidate eigenvalue of a is the smallest size of perturbation a such that z is an exact eigenvalue of a a 2 the pseudospectrum associated with a backward error of level is a z c z it contains all the points z which are seen as eigenvalues within an accuracy tolerance of in this paper we focus on normwise and homotopic perturbations which yield respectively for the approximate eigenvalue the backward errors 1 a a i 1 and e a e a i 1 an application to the arnoldi method is presented

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

realization extension and classification of certain physically important groups and algebras an associative algebra of differential forms with division has been constructed the algebra of forms in each different space provides a practical realization of the universal clifford algebra of that space a classification of all such algebras is given in terms of two distinct types of algebras nk and sk the former include the dihedral quaternion and majorana algebras the latter include the complex spinor and dirac algebras the associative product expresses hodge duality as multiplication by a basis element this makes possible the realization of higher order algebras in a calculationally useful algebraic setting the fact that the associative algebras as well as the enveloped lie algebras are precisely those arising in physics suggests that this formalism may be a convenient setting for the formulation of basic physical laws

stability of eigenvalues and spectral decompositions under linear perturbation abstract given an n by n matrix a and a fixed perturbation matrix e the effect of the linear perturbations a te t c on certain spectral properties of a is analyzed specially the effect of the structure of e on these properties is investigated a geometric framework for spectral analysis of a te is developed to achieve this goal which also leads to a better understanding of the sensitivity of eigenvalues and spectral decompositions of a

free divisors in prehomogeneous vector spaces we study linear free divisors that is free divisors arising as discriminants in prehomogeneous vector spaces and in particular in quiver representation spaces we give a characterization of the prehomogeneous vector spaces containing such linear free divisors for reductive linear free divisors we prove that the numbers of geometric and representation theoretic irreducible components coincide as a consequence we find that a quiver can only give rise to a linear free divisor if it has no oriented or unoriented cycles we also deduce that the linear free divisors which appear in sato and kimura s list of irreducible prehomogeneous vector spaces are the only irreducible reductive linear free divisors furthermore we show that all quiver linear free divisors are strongly euler homogeneous that they are locally weakly quasihomogeneous at points whose corresponding representation is not regular and that all tame quiver linear free divisors are locally weakly quasihomogeneous in particular the latter satisfy the logarithmic comparison theorem

dynamical invariance algebra of the hartmann potential the accidental degeneracy occurring in the quantum mechanical treatment of the ring shaped potential v eta sigma 2r 1 1 2q eta 2 sigma 2 r sin theta 2 is explained by constructing an su 2 dynamical invariance algebra the schrodinger equation is solved in parabolic coordinates written in the framework of the kustaanheimo stiefel transformation and the hamilton jacobi equations are solved in ordinary parabolic coordinates all finite trajectories are found to be periodic

complementarity relation between the u p q and u n lie groups and some applications to atomic physics the complementarity relation between the unitary groups u d and u n within the symmetrical irreducible representations of the larger unitary group u dn is extended to non compact groups it is proved that the pseudo unitary group u p q is complementary with respect to u n within some positive discrete series irreducible representations of the larger pseudo unitary group u pn qn the latter arise when reducing the metaplectic irreducible representations 1 2 dn and 1 2 dn 13 2 of the real symplectic group sp 2dn r where d p q and they are characterised by a single label the eigenvalue of the first order casimir operator some applications of the u p q u n complementarity to atomic physics are outlined for such purposes the isomorphism between the lie algebras of su 2 2 and so 4 2 is used extensively the mathematical framework underlying the kibler negadi approach 1983 4 of the hydrogen atom dynamical group is extended to the independent electron dynamical group of intrashell many electron states as well as to the correlated electron dynamical group of intrashell doubly excited states

kepler orbits and the harmonic oscillator it is shown that the kustaanheimo stiefel transformation which transforms the three dimensional kepler problem into that for a four dimensional harmonic oscillator may be expressed in terms of quaternions in this form the transformation represents a continuous mapping of a triad of orthonormal vectors fixed in space into a rotating triad of orthogonal vectors where one of the unit vectors is mapped into the position vector of a moving particle the reduction of the kepler problem to that of a harmonic oscillator is straightforward and direct in the quaternion formalism the complex stereographic transformation used recently by the author in the corresponding reduction of schrodinger s equation for the hydrogen atom is shown to be closely related to the quaternion representation

effect of linear perturbation on spectra of matrices abstract given n by n matrices a and e a characterization of the sensitivity of eigenvalues of a with respect to the perturbation a te is provided distribution of eigenvalues of the family a te t c in the complex plane is analyzed and necessary and sufficient conditions are provided for which a and a te are isospectral for all t c

treatment of some singular potentials by change of variables in wiener integrals using a regularization of the classical kepler problem we show that a change of process and a change of time in stochastic differential equations allow a treatment of singular potentials in wiener integrals

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

a general method of generating and classifying clifford algebras a general construction program is given for the generation of higher order clifford algebras of various signatures together with their faithful representations in terms of pauli type operators the analysis is based upon several isomorphism theorems that provide a simpler understanding of the standard classification scheme of these algebras and an improved signature index notation is suggested that identifies each of the five types of clifford algebras

cohomology of the complement of a free divisor we prove that if d is a strongly quasihomogeneous free divisor in the stein manifold x and u is its complement then the de rham cohomology of u can be computed as the cohomology of the complex of meromorphic differential forms on x with logarithmic poles along d with exterior derivative the class of strongly quasihomogeneous free divisors introduced here includes free hyperplane arrangements and the discriminants of stable mappings in mather s nice dimensions and in particular the discriminants of coxeter groups

theory of nonbijective canonical transformations in mechanics application to the coulomb problem it is shown that the study of nonbijective transformation requires a fiber bundle formulation of mechanics the conditions upon which nonbijective canonical point transformations can be defined are given then as an example we apply that theory to the study of the coulomb problem in two and three dimensions the hopf fibration leads to an inverse harmonic oscillator problem since the completion of this work a paper by g h ringwood and j t devreese has been published in j math phys 21 1390 1980 dealing with the same problem their work is based on the construction of propagators in quotient spaces the identity between the propagator prescriptions and nonlinear canonical transformations is not automatically fulfilled therefore it seems that the reliability of their results is not due to their method which in general is not correct but to an underlying property of the transformation used namely the kustaanheimo stiefel map see our results

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

the le varieties ii for a complex polynomial f n 1 0 0 with a singular set of complex dimensions at the origin we define a sequence of varieties the le varieties off at 0 the multiplicities of these varieties generalize the milnor number for an isolated singularity in particular we show that ifs n 2 the milnor fibre off is obtained fromb 2n by successively attaching k handles wheren s k n ifs n 1 the milnor fibre off is obtained from a2n manifold with the homotopy type of a bouquet of circles by successively attaching k handles where 2 k n

answers to some equisingularity questions we give examples of families of hypersurface singularities with constant le numbers constant milnor fibration and non constant topological type answering negatively a question of d massey on the other hand we prove that the constancy of the le numbers implies that the homotopy type of the link is constant along the family as an application we give an example of a flat family of projective reduced and irreducible hypersurfaces having the same homotopy type but different topological type another application is an example of a whitney trivial family of isolated singularities such that the topological type of the projectivised tangent cone is not constant in the family the last example answers negatively a question of o zariski

algebras with three anticommuting elements i spinors and quaternions a general construction of alternative algebras with three anticommuting elements and a unit is given as an exhaustive result over the real and complex fields we obtain the clifford algebras h quaternions n1 dihedral clifford algebra which is related to real 2 spinors and s1 algebra of pauli matrices which is related to complex 2 spinors what is important is that the algebras n1 and s1 possess inverses everywhere except on a region akin to the light cone of the minkowski space while the quaternion algebra h has inverses everywhere except at the zero element we discuss the reasons why the three algebras n1 h and s1 are so difficult to distinguish in the representation space of 2 2 complex matrices

a construction relating clifford algebras and cayley dickson algebras a review of the applications of the octonions in physics is given a construction is presented both the cayley dickson algebras and the clifford algebras arise naturally under this construction from the quaternion algebras the mathematical properties of the algebras constructed are discussed

the hurwitz transformation nonbilinear version an alternative approach to the hurwitz h transformation reducing euclidean space e8 to euclidean space e5 is developed it is shown how refusal of the bilinearity condition leads to the replacement of the h transformation by its modified nonbilinear left and right version hl and hr transformations the hl hr transformation of the specific type eight dimensional hyperspherical coordinates is investigated the radial coordinate u and the polar angle 2 in this approach transform into u2 and respectively like in the h case action of these transformations on the remaining hyperspherical coordinates unlike in the h case is equivalent to the invariance shutting of one angular triplet and the shutting invariance of another triplet the connection of hl and hr with h is established and the structure of the h transformation itself is revealed on this basis

pseudozeros of polynomials and pseudospectra of companion matrices it is well known that the zeros of a polynomial p are equal to the eigenvalues of the associated companion matrix a in this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding the epsilon pseudozero set z epsilon p is the set of zeros of all polynomials hat p obtained by coefficientwise perturbations of p of size leq epsilon this is a subset of the complex plane considered earlier by mosier and is bounded by a certain generalized lemniscate the epsilon pseudospectrum lambda epsilon a is another subset of bbb c defined as the set of eigenvalues of matrices hat a a e with vert e vert leq epsilon it is bounded by a level curve of the resolvent of a we find that if a is first balanced in the usual eispack sense then z epsilon vert p vert p and lambda epsilon vert a vert a are usually quite close to one another it follows that the matlab roots algorithm of balancing the companion matrix then computing its eigenvalues is a stable algorithm for polynomial zerofinding experimental comparisons with the jenkins traub imsl and madsen reid harwell fortran codes confirm that these three algorithms have roughly similar stability properties

the hurwitz transformation and generalized cayley klein parameterization the fiber bundles and global rotations arising in the kustaanheimo stiefel and hurwitz transformations are studied generalized cayley klein parameterization is introduced in a functional form and is used for the analysis of fibration in hurwitz transformations essential differences in fibration structure between these transformations similar in cartesian description are established in the angular map

generalized quaternions and spacetime symmetries the construction of a class of associative composition algebras qn on r4 generalizing the well known quaternions q provides an explicit representation of the universal enveloping algebra of the real three dimensional lie algebras having tracefree adjoint representations class a bianchi type lie algebras the identity components of the four dimensional lie groups gl qn 1 qn general linear group in one generalized quaternion dimension which are generated by the lie algebra of this class of quaternion algebras are diffeomorphic to the manifolds of spacetime homogeneous and spatially homogeneous spacetimes having simply transitive homogeneity isometry groups with tracefree lie algebra adjoint representations in almost all cases the complete group of isometries of such a spacetime is isomorphic to a subgroup of the group of left and right translations and automorphisms of the appropriate generalized quaternion algebra similar results hold for the single class b lie algebra of bianchi type v characteri

algebras with anticommuting basal elements space time symmetries and quantum theory a collection a of algebras with anticommuting basal elements is investigated it is shown that the collection a includes the quaternions the octonions the algebra of color as well as other algebras familiar to the physicist each algebra a is a quadratic jordan admissible algebra and possesses a norm that is a generalization of the minkowski metric using a cayley dickson like process each such algebra a in a can be embedded into a larger algebra a that is also in the collection a these algebras should provide candidates for models to describe observables color and other phenomena encountered in particle physics

a remarkable spectral feature of the schrodinger hamiltonian of the harmonic oscillator perturbed by an attractive δ interaction centred at the origin double degeneracy and level crossing we rigorously define the self adjoint hamiltonian of the harmonic oscillator perturbed by an attractive interaction of strength centred at 0 the bottom of the confining parabolic potential by explicitly providing its resolvent our approach is based on a coupling constant renormalization related to a technique originated in quantum field theory and implemented in the rigorous mathematical construction of the self adjoint operator representing the negative laplacian perturbed by the interaction in two and three dimensions the way the interaction enters in our hamiltonian corresponds to the one originally discussed for the free hamiltonian instead of the harmonic oscillator one by p s ba it should not be confused with the potential perturbation of the harmonic oscillator discussed e g in a recent paper by gadella glasser and nieto also introduced by p s ba as a perturbation of the one dimensional free laplacian and recently investigated in that context by golovaty hryniv and zolotaryuk we investigate in detail the spectrum of our perturbed harmonic oscillator the spectral structure differs from that of the one dimensional harmonic oscillator perturbed by an attractive interaction centred at the origin the even eigenvalues are not modified at all by the interaction moreover all the odd eigenvalues regarded as functions of exhibit the rather remarkable phenomenon called level crossing after first producing the double degeneracy of all the even eigenvalues for the value b being the beta function

on the connection between the hydrogen atom and the harmonic oscillator the continuum case the connection between a three dimensional nonrelativistic hydrogen atom with positive energy and a four dimensional isotropic harmonic oscillator with repulsive potential is established by applying jordan schwinger boson calculus to the algebra of the laplace runge lenz pauli vector the spectrum generating group so 4 2 both for the bound and free states of the three dimensional hydrogen atom arises as a quotient of the group sp 8 r associated to four dimensional isotropic harmonic oscillator with constraint

non relativistic coulomb problem in a one dimensional quantum mechanics it is shown that unlike the model based only on the potential singularity u x mod x mod 1 the group of hidden symmetry explains not only the double degeneration of the energy spectrum but also the explicit form of the spectrum wavefunctions and an extra constant of motion analogous to the runge lenz vector

algebras with three anticommuting elements ii two algebras over a singular field the clifford algebra generated by the elements 1 with 2 1 is an abelian ring of dimension two with properties analogous to the complex field c the ring has a string of singular inverses and may be regarded as a singular field which circumvents both the fundamental theorem of algebra and the frobenius theorem we construct two associative algebras of dimension four over the clifford algebra 1 and the biquaternions of clifford 2 and demonstrate that both algebras possess inverses everywhere except on a singular region akin to the light cone of the minkowski space matrix representations are discussed as well as the importance of the algebras 1 and 2 in the description of physical vector fields

topological equisingularity of hypersurfaces with 1 dimensional critical set abstract we focus on topological equisingularity of families of holomorphic function germs with 1 dimensional critical set we introduce the notion of equisingularity at the critical set and prove that any family which is equisingular at the critical set is topologically equisingular we show that if a family of germs with 1 dimensional critical set has constant generic le numbers then it is equisingular at the critical set and hence topologically equisingular answering a question of d massey 13 it is worth to remark that this does not happen for higher dimensional critical set 5 we use these topological triviality results to modify the definition of singularity stem present in the literature introducing and characterising topological stems being this concept closely related with arnold s series of singularities we provide another sufficient condition for topological equisingularity for families whose reduced critical set is deformed flatly finally we study how the critical set can be deformed in a topologically equisingular family and provide examples of topologically equisingular families whose critical set is a non flat deformation with singular special fibre and smooth generic fibre

characterizing normal crossing hypersurfaces the objective of this article is to give an effective algebraic characterization of normal crossing hypersurfaces in complex manifolds it is shown that a divisor hypersurface has normal crossings if and only if it is a free divisor has a radical jacobian ideal and a smooth normalization using k saito s theory of free divisors also a characterization in terms of logarithmic differential forms and vector fields is found finally we give another description of a normal crossing divisor in terms of the logarithmic residue using recent results of m granger and m schulze

neighborhoods of algebraic sets in differential topology a smooth submanifold in a manifold has a tubular neighborhood and in piecewise linear topology a subcomplex of a simplicial complex has a regular neighborhood the purpose of this paper is to develop a similar theory for algebraic and semialgebraic sets the neighbor hoods will be defined as level sets of polynomial or semialgebraic functions introduction let m be an algebraic set in real n space r and let x be a compact algebraic subset of m containing its singular locus if any an algebraic neighborhood of a in m is defined to be a 1 0 8 where o 0 is sufficiently small and a m r is a proper polynomial function for which a 0 and a 1 0 x such an a will be called a rug function occasionally we will need rational or analytic a but this is not a significant generalization algebraic neighborhoods always exist the curve selec tion lemma is used to prove uniqueness anyone familiar with milnor 2 will recognize the technique since uniqueness is a crucial result and since there are a few trouble some small points this proof is given in detail the uniqueness theorem shows that the link of a singularity of an algebraic set m is independent of the embedding of m in its ambient space i have been unable to find a proof of this result in the litera ture in addition an algebraic neighborhood of a nonsingular x in m is shown to be a tubular neighborhood in the sense of differential topology this material is in 1 in 2 the theory is rapidly developed for real and complex projective space by embedding these spaces in real affine space as an application it is shown that when m is an affine algebraic set with projective completion m then the complement in m of a large ball centered at the origin of affine space is an alge braic neighborhood of the intersection of m with the hyperplane at infinity in 3 the theory is generalized to the case where m is a semialgebraic subset of r and x is a compact semialgebraic subset of m for example x could be a non isolated singular point of m a semialgebraic neighborhood of x in m is defined to be a 1 0 o where o 0 is sufficiently small and a m r is a proper semialgebraic function for which a 0 and a 1 0 x again these neighborhoods exist and are unique this is shown by mimicking the proof in the algebraic case using semialgebraic

pseudo euclidean hurwitz pairs sigma and pure spinor models considers the pseudo euclidean hurwitz pairs as formulated in a recent study by randriamihamison 1990 from these pairs the author constructs solutions of sigma models defined on real hyperquadrics and solutions of pure spinor models this provides a non trivial extension of previous works of fujii 1988