topological tensor current of p branes in the φ mapping theory we present a new topological tensor current of p branes by making use of the mapping theory it is shown that the current is identically conserved and behaves as and every isolated zero of the vector field x corresponds to a magnetic p brane using this topological current the generalized nambu action for multi p branes is given and the field strength f corresponding to this topological tensor current is obtained it is also shown that the magnetic charges carried by p branes are topologically quantized and labeled by hopf index and brouwer degree the winding number of the mapping

quantum field theory and hopf algebra cohomology we exhibit a hopf superalgebra structure of the algebra of field operators of quantum field theory qft with the normal product based on this we construct the operator product and the time ordered product as a twist deformation in the sense of drinfeld our approach yields formulae for perturbative products and expectation values that allow for a significant enhancement in computational efficiency as compared to traditional methods employing hopf algebra cohomology sheds new light on the structure of qft and allows the extension to interacting not necessarily perturbative qft we give a reconstruction theorem for time ordered products in the spirit of streater and wightman and recover the distinction between free and interacting theory from a property of the underlying cocycle we also demonstrate how non trivial vacua are described in our approach solving a problem in quantum chemistry

on interval enclosures using slope arithmetic this paper describes how a combination of interval derivative arithmetic and interval slope arithmetic can be used with the programming language pascal sc to obtain an enclosure of the range of a factorable function r n r 1 with c 2 d which is often narrower than that which is obtained by using slope arithmetic alone and which is always narrower than that which is obtained by using interval derivative arithmetic alone

on an easy transition from operator dynamics to generating functionals by clifford algebras clifford geometric algebras of multivectors are treated in detail these algebras are built over a graded space and exhibit a grading or multivector structure the careful study of the endomorphisms of this space makes it clear that opposite clifford algebras have to be used for obtaining all endomorphisms based on these mathematics we give a fully clifford algebraic account on generating functionals which is thereby geometric the field operators are shown to be clifford and opposite clifford maps this picture relying on geometry does not need positivity in principle furthermore we propose a transition from operator dynamics to generating functionals which is based on the algebraic techniques as a calculational benefit this transition is considerably short compared to standard ones the transition is not injective unique and depends additionally on the choice of an ordering we obtain a direct and constructive connection between orderings and the explicit form of the functional hamiltonian these orderings depend on the propagator of the theory and thus on the ground state this is invisible in path integral formulations the method is demonstrated within two examples a nonlinear spinor field theory and spinor qed antisymmetrized and normal ordered functional equations are derived in both cases

on the classification of clifford algebras and their relation to spinors in n dimensions a classification of all the clifford algebras is given in terms of kronecker products of the quaternion and dihedral groups the relationship to spinors in n dimensions is explicitly determined we show that the real clifford algebra in minkowski spacetime is distinct from both the algebra of dirac matrices and the algebra of majorana matrices and cannot be realized by the spinor framework the matrix representations of clifford algebras are discussed and are utilized to give a classification of the real forms of lie algebras we are thus able to relate clifford lie and spinor algebras in an intrinsic geometrical setting

medical image segmentation volume representation and registration using spheres in the geometric algebra framework this paper presents an algorithm to model volumetric data and other one for non rigid registration of such models using spheres formulated in the geometric algebra framework the proposed algorithm for modeling as opposite to the union of spheres method reduces the number of entities spheres used to model 3d data our proposal is based in marching cubes idea using however spheres while the union of spheres uses delaunay tetrahedrization the non rigid registration is accomplished in a deterministic annealing scheme at the preprocessing stage we segment the objects of interest by a segmentation method based on texture information this method is embedded in a region growing scheme as our final application we present a scheme for surgical object tracking using again geometric algebra techniques

an unconditionally convergent method for computing zeros of splines and polynomials we present a simple and ecient method for computing zeros of spline functions the method exploits the close relationship between a spline and its control polygon and is based on repeated knot insertion like newton s method it is quadratically convergent but the new method overcomes the principal problem with newton s method in that it always converges and no starting value needs to be supplied by the user

supersymmetry and clifford algebras a canonical derivation of extended lie superalgebras an p q right from the metric structure of the underlying space time e p q is presented the canonical method is that of clifford algebras c p q and their representations theory the results are tabularized

a hopf laboratory for symmetric functions an analysis of symmetric function theory is given from the perspective of the underlying hopf and bi algebraic structures these are presented explicitly in terms of standard symmetric function operations particular attention is focused on laplace pairing sweedler cohomology for 1 and 2 cochains and twisted products rota cliffordizations induced by branching operators in the symmetric function context the latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type a commentary on related issues in the combinatorial approach to quantum field theory is given

bilinear covariants and spinor fields duality in quantum clifford algebras classification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum clifford algebras on minkowski spacetime once the bilinear covariants are expressed in terms of algebraic spinor fields the duality between spinor and quantum spinor fields can be discussed thus by endowing the underlying spacetime with an arbitrary bilinear form with an antisymmetric part in addition to a symmetric spacetime metric quantum algebraic spinor fields and deformed bilinear covariants can be constructed they are thus compared to the classical non quantum ones classes of quantum spinor fields classes are introduced and compared with lounesto s spinor field classification a physical interpretation of the deformed parts and the underlying z grading is proposed the existence of an arbitrary bilinear form endowing the spacetime already has been explored in the literature in the context of quantum gravity s w hawking the unpredictability of quantum gravity commun

real representations of finite clifford algebras i classification a classification of real matrix irreducible representations of finite dimensional real clifford algebras has been made in contrast to the case of complex representation three distinct types of representations can be obtained which we call normal almost complex and quaternionic the dimension of the latter two cases is twice as large as that of the normal representation a criteria for a given clifford algebra to possess a particular type of the representations is also given with some applications

topological structure of gauss bonnet chern density and its topological current the decomposition of spin connection on the sphere bundle of a compact and oriented even dimensional riemannian manifold is detailed it is shown that the gauss bonnet chern form can be expressed in terms of a smooth vector field and taken the form of the function the topological current of the gauss bonnet chern theorem and its topological structure are discussed

multilinear representations of rotation groups within geometric algebra it is shown that higher weighted representations of rotation groups can be constructed using multilinear functions in geometric algebra methods for obtaining the irreducible representations are found and applied to the spatial rotation group so 3 and the proper lorentz group so 1 3 it is also shown that the representations can be generalized to non linear functions with applications to relativistic wave equations describing higher spin particles such as the rarita schwinger equations the internal spin degrees of freedom and the external space time degrees of freedom are handled within the same mathematical structure

subdivision algorithms converge quadratically this paper is concerned with determining the exact convergence rates of subdivision algorithms for various settings of polynomial and spline curve or surface representations

near minimally normed spline quasi interpolants on uniform partitions spline quasi interpolants qis are local approximating operators for functions or discrete data we consider the construction of discrete and integral spline qis on uniform partitions of the real line having small infinity norms we call them near minimally normed qis they are exact on polynomial spaces and minimize a simple upper bound of their infinity norms we give precise results for cubic and quintic qis also the qi error is considered as well as the advantage that these qis present when approximating functions with isolated discontinuities

inner topological structure of hopf invariant in light of mapping topological current theory the inner topological structure of hopf invariant is investigated it is revealed that hopf invariant is just the winding number of gauss mapping according to the inner structure of topological current a precise expression for hopf invariant is also presented it is the total sum of all the self linking and all the linking numbers of the knot family

the topological structure of nieh yan form and the chiral anomaly in spaces with torsion the topological structure of the nieh yan form in a four dimensional manifold is given by making use of the decomposition of spin connection the case of the generalized nieh yan form on a 2d dimensional manifold is discussed with an example of an eight dimensional case which is studied in detail the chiral anomaly with nonvanishing torsion is also studied the further contributions from the torsional part to the chiral anomaly are found to estimate from the zeros of some fields under the pure gauge condition

new geometric methods for computer vision an application tostructure and motion estimation we discuss a coordinate free approach to the geometry of computer vision problems the technique we use to analyse the three dimensional transformations involved will be that of geometric algebra a framework based on the algebras of clifford and grassmann this is not a system designed specifically for the task in hand but rather a framework for all mathematical physics central to the power of this approach is the way in which the formalism deals with rotations for example if we have two arbitrary sets of vectors known to be related via a 3d rotation the rotation is easily recoverable if the vectors are given extracting the rotation by conventional means is not as straightforward the calculus associated with geometric algebra is particularly powerful enabling one in a very natural way to take derivatives with respect to any multivector general element of the algebra what this means in practice is that we can minimize with respect to rotors representing rotations vectors representing translations or any other relevant geometric quantity this has important implications for many of the least squares problems in computer vision where one attempts to find optimal rotations translations etc given observed vector quantities we will illustrate this by analysing the problem of estimating motion from a pair of images looking particularly at the more difficult case in which we have available only 2d information and no information on range while this problem has already been much discussed in the literature we believe the present formulation to be the only one in which least squares estimates of the motion and structure are derived simultaneously using analytic derivatives

computing the range of values of real functions using b spline form abstract in this paper we present an easy and efficient method for computing the range of a function by using spline quasi interpolation we exploit the close relationship between the spline function and its control polygon and use tight subdivision technique in order to obtain monotonic splines which make the range of the spline easy to compute the proposed method is useful in case of given scattered data generated by some unknown function f or scientific measurements several numerical examples are given for cubic and quintic quasi interpolant approximant to illustrate the efficiency and the performance of our method

a grassmann integral equation the present study introduces and investigates a new type of equation which is called grassmann integral equation in analogy to integral equations studied in real analysis a grassmann integral equation is an equation which involves grassmann berezin integrations and which is to be obeyed by an unknown function over a finite dimensional grassmann algebra gm i e a sought after element of the grassmann algebra gm a particular type of grassmann integral equations is explicitly studied for certain low dimensional grassmann algebras the choice of the equation under investigation is motivated by the effective action formalism of lattice quantum field theory in a very general setting for the grassmann algebras g2n n 2 3 4 the finite dimensional analogues of the generating functionals of the green functions are worked out explicitly by solving a coupled system of nonlinear matrix equations finally by imposing the condition g g0 const 0 r k k k 1 n are the generat

vertex normal ordering as a consequence of nonsymmetric bilinear forms in clifford algebras we consider clifford algebras with nonsymmetric bilinear forms they parametrize the chosen ideal in the isomorphism class of the standard symmetric ones since the content of physical theories depends on the injection n nv cl v q one has to transform to the standard construction the injection is described by the antisymmetric part of the bilinear form this process results in the appropriate vertex normal ordering terms which are now obtained from the theory itself and not added ad hoc via a regularization argument

on the hopf algebraic origin of wick normal ordering a combinatorial formula of rota and stein is taken to perform wick reordering in quantum field theory wick s theorem becomes a hopf algebraic identity called cliffordization the combinatorial method relying on hopf algebras is highly efficient in computations and yields closed algebraic expressions

computing slope enclosures by exploiting a unique point of inflection using slope enclosures may provide sharper bounds for the range of a function than using enclosures of the derivative hence slope enclosures may be useful in verifying the assumptions for existence tests or in algorithms for global optimization previous papers by kolev and ratz show how to compute slope enclosures for convex and concave functions in this paper we generalize these formulas and show how to obtain slope enclosures for a function that has exactly one point of inflection or whose derivative has exactly one point of inflection

inclusion functions and global optimization 11 inclusion functions combined with special subdivision strategies are an effective means of solving the global unconstrained optimization problem although these techniques were determined and numerically tested about ten years ago they are nearly unknown and scarcely used in order to make the role of inclusion functions and subdivision strategies more widespread and transparent we will discuss a related simplified basic algorithm it computes approximations of the global minimum and at the same time bounds the absolute approximation error we will show that the algorithm works and converges under more general assumptions than it has been known hitherto that is only appropriate inclusion functions are expected to exist the number of minimal points finite or infinite is not of importance lipschitz conditions or continuity are not assumed as shown in the appendix the required inclusion functions can be constructed and programmed without difficulty in a natural way using interval analysis

a new pruning test for finding all global minimizers of nonsmooth functions a new pruning test is presented via interval slopes for finding all global minimizers of nonsmooth functions of several variables based on an expansion scheme which can be used to achieve better interval slopes and enclosure of function ranges the presented pruning test extends ratz s a nonsmooth global optimization technique using slopes one dimensional case j global optimiz 4 1999 365 393 pruning technique which can be extensively applied in several variable nonsmooth global optimization as a accelerating device this new pruning test is similar to the monotonicity test frequently used in interval methods for smooth problems but it improves the monotonicity test numerical results show that the proposed global optimization algorithm with the new pruning test is better than that with the monotonicity test for smooth functions and superior to other interval algorithm for nonsmooth functions

a new multi section based technique for constrained optimization problems with interval valued objective function in this paper an efficient optimization technique is proposed for constrained optimization problems with interval valued objective function at first the significance of interval valued objective function and the meaning of the interval valued solution of the proposed problem have been explained with graphical interpretation generally this type of problems has infinitely many compromise solutions the aim of this approach is to obtain one of such solutions with higher accuracy and lower computational cost the proposed technique is mainly based on the splitting criterion of the accepted prescribed search region calculation of the interval inclusion functions and the selection of subregion depending on the modified interval order relations in the context of the decision makers point of view novel interval oriented constraint satisfaction rules are used for non interval equality and inequality constraints clearly the proposed technique is nothing but an imitation of well known interval analysis based branch and bound b b optimization approach the modified multi section division criterion with some new interval oriented constraint satisfaction rules for non interval valued equality and inequality constraints and some novel interval order relations in the context of the decision makers point of view have been applied to increase the efficiency of the proposed algorithm finally the technique is applied for solving some test problems and the results are compared with the same obtained from the existing methods

a geometric approach for the analysis and computation of the intrinsic camera parameters the authors of this paper adopted the projected characteristics of the absolute conic in terms of the pascal s theorem to propose an entirely new camera calibration method based on purely geometric thoughts the use of this theorem in the geometric algebra framework allows us to compute a projective invariant using the conics of only two images which expressed using brackets helps us to set enough equations to solve the calibration problem the method requires restricted controlled camera movements our method is less sensitive to noise as the kruppa s equation based methods experiments with simulated and real images con rm that the performance of the algorithm is reliable 2001pattern recognition society published by elsevier science ltd all rights reserved

supersymmetric yang mills theory on a four manifold by exploiting standard facts about n 1 and n 2 supersymmetric yang mills theory the donaldson invariants of four manifolds that admit a kahler metric can be computed the results are in agreement with available mathematical computations and provide a powerful check on the standard claims about supersymmetric yang mills theory

calculus on the multivector manifold spin p q coordinate free differential and integral calculus is formulated on the spin groups regarded as multivector manifolds embedded in geometric algebra the derivative with respect to a point on the manifold is shown to be in the tangent algebra at the inverse point the curvature of spin p q is calculated and related to the structure constants of its lie algebra volume elements on spin p q are introduced naturally as completely antisymmetrized geometric products of bivectors they are left invariant like haar measure and are elements of a multivector valued grassman algebra it is shown that integration in this sense is forbidden for many spin groups a version of stokes s theorem is formulated on spin p q

berezin integral as a limit of riemann sum berezin integration of functions of anticommuting grassmann variables is usually seen as a formal operation sometimes even defined via differentiation using the formalism of geometric algebra and geometric calculus in which the grassmann numbers are endowed with a second associative product coming from a clifford algebra structure we show how berezin integrals can be realized in the high dimensional limit as integrals in the sense of geometric calculus we then show how the concepts of spinors and superspace transform into this framework

a search for chaotic behavior in large and mesoscale motions in the pacific ocean the lagrangian trajectories of three satellite tracked bouys deployed in the pacific ocean for a one year period are studied from the standpoint of dynamical systems theory time series of the bouy positions are analyzed to estimate the correlation dimension and kolmogorov entropy we find a small finite value of correlation dimension near 1 4 and a finite kolmogorov entropy of about 0 2 day 1 these results at first suggest that the bouy motions may be dominated by a low dimensional chaotic attractor using standard ideas from the mathematics of fractals and a new method which we call multivariate scaling analysis we are able to demonstrate however that the bouy trajectories are instead fractal curves in the physical plane x y these results may be viewed from the standpoint of multivariate nongaussian convergence of the scales which occur when the fourier spectrum is described by a power law such systems can have a small finite value for the correlation dimension and are therefore fractal even when no underlying attractor is present systems of this type are more easily related to turbulent behavior stochastic many degrees of freedom rather than to strange attractor deterministic few degrees of freedom

the general decomposition theory of su 2 gauge potential topological structure and bifurcation of su 2 chern density by means of the geometric algebra the general decomposition of su 2 gauge potential on the sphere bundle of a compact and oriented four dimensional manifold is given using this decomposition theory the su 2 chern density has been studied in detail it shows that the su 2 chern density can be expressed in terms of the function one can also find that the zero points of the vector fields are essential to the topological properties of a manifold it is shown that there exists the crucial case of branch process at the zero points based on the implicit function theorem and the taylor expansion the bifurcation of the chern density is detailed in the neighborhoods of the bifurcation points of it is pointed out that since the chern density is a topological invariant the sum topological chargers of the branches will remain constant during the bifurcation process

dynamics and control of the 2 d navier stokes equations abstract this paper deals with the dynamics and control of the two dimensional 2 d navier stokes n s equations with a spatially periodic and temporally steady forcing term first we construct a dynamical system of nine nonlinear differential equations by fourier expansion and truncation of the 2 d n s equations then we study the dynamics of the obtained reduced order system by analyzing the system s attractors for different values of the reynolds number r e by applying the symmetry of the equations on one of the system s attractors a symmetric limit trajectory that is part of the dynamics is obtained moreover a lyapunov based control strategy to control the dynamics of the system for a given r e is designed finally numerical simulations are undertaken to validate the theoretical developments

passive scalars three dimensional volume preserving maps and chaos the dynamics of a medium sized particle passive scalar suspended in a general time periodic incompressible fluid flow can be described by three dimensional volume preserving maps in this paper these maps are studied in limiting cases in which some of the variables change very little in each iteration and others change quite a lot the former are called slow variables or actions the latter fast variables or angles the maps are classified by their number of actions for maps with only one action we find strong evidence for the existence of invariant surfaces that survive the nonlinear perturbation in a kam like way on the other hand for the two action case the motion is confined to invariant lines that break for arbitrary small size of the nonlinearity instead we find that adiabatic invariant surfaces emerge and typically intersect the resonance sheet of the fast motion at these intersections surfaces are locally broken and transitions from one to another can occur we call this process which is analogous to arnold diffusion singularity induced diffusion it is characteristic of two action maps in one action maps this diffusion is blocked by kam like surfaces

centered forms for functions in several variables abstract r e moore 3 has introduced the centered form of a rational function f for obtaining good and easily computable approximations which include the exact range of f over an interval x moore s definition is implicit while explicit formulas for the centered form are given in 8 and 9 in 9 centered forms of higher order for functions of one variable are developed which lead to better estimations than the centered forms defined originally in this paper centered forms of higher order for rational functions in several variables are explicitly defined they also give better approximations than the centered forms defined originally

triad angle locking in low order models of the 2d navier stokes equations abstract in recent years franceschini and his colleagues have explored the detailed trajectory behavior of several low order truncated models of the 2d navier stokes equations their models however involve only the real amplitude the real amplitude only the excitation is through the third mode only to simulate the 2d flow more accurately we examined a sequency of low order models which though truncated a la the pattern of franceschini et al have complex amplitude modes as dictated by the navier stokes equations a surprise was that the complex mode models have a special equilibrium state for triad angle sums being locked to 2 3 2 the upshot is then the manifold of complex mode models collapses to a half sized manifold of the corresponding real models we have traced this to the single mode excitation hence the triad angle looking can be avoided by forcing several modes

quantum clifford algebra from classical differential geometry we show the emergence of clifford algebras of nonsymmetric bilinear forms as cotangent algebras of kaluza klein kk spaces pertaining to teleparallel space times these spaces are canonically determined by the horizontal differential invariants of finsler bundles of the type b m s m where b m is the set of all the tangent frames to a differentiable manifold m and where s m is the sphere bundle if m is space time itself m4 the geometric phase space s m4 has dimension seven this reformulation of the horizontal invariants as pertaining to a kk space removes the mismatch between the dimensionality of the tangent frames to m4 and the dimensionality of s m4 in the kk space a symmetric tangent metric induces a cotangent metric which is not symmetric in general an interior covariant derivative in the sense of kaehler is defined it involves the antisymmetric part of the cotangent metric which thus enters electrodynamics and the dirac equation

interval approximation of higher order to the ranges of functions the bernstein and b spline forms are generalized to multivariate polynomials these forms are combined with a type of taylor form for multivariate functions to generate realizable forms for multivariate functions

algebraic and dirac hestenes spinors and spinor fields almost all presentations of dirac theory in first or second quantization in physics and mathematics textbooks make use of covariant dirac spinor fields an exception is the presentation of that theory first quantization offered originally by hestenes and now used by many authors there a new concept of spinor field as a sum of nonhomogeneous even multivectors fields is used however a careful analysis detailed below shows that the original hestenes definition cannot be correct since it conflicts with the meaning of the fierz identities in this paper we start a program dedicated to the examination of the mathematical and physical basis for a comprehensive definition of the objects used by hestenes in order to do that we give a preliminary definition of algebraic spinor fields asf and dirac hestenes spinor fields dhsf on minkowski space time as some equivalence classes of pairs u u where u is a spinorial frame field and u is an appropriate sum of multivectors fields to be specified below the necessity of our definitions are shown by a careful analysis of possible formulations of dirac theory and the meaning of the set of fierz identities associated with the bilinear covariants on minkowski space time made with asf or dhsf we believe that the present paper clarifies some misunderstandings past and recent appearing on the literature of the subject it will be followed by a sequel paper where definitive definitions of asf and dhsf are given as appropriate sections of a vector bundle called the left spin clifford bundle the bundle formulation is essential in order to be possible to produce a coherent theory for the covariant derivatives of these fields on arbitrary riemann cartan space times the present paper contains also appendixes a e which exhibits a truly useful collection of results concerning the theory of clifford algebras including many tricks of the trade necessary for the intelligibility of the text

a seven mode truncation of the plane incompressible navier stokes equations a model obtained by a seven mode truncation of the navier stokes equations for a two dimensional incompressible fluid on a torus is studied this model extending a previously studied five mode one exhibits a very rich and varied phenomenology including some remarkable properties of hysteresis i e coexistence of attractors a stochastic behavior is found for high values of the reynolds number when no stable fixed points closed orbits or tori are present

z2 gradings of clifford algebras and multivector structures let c v g be the real clifford algebra associated with the real vector space v e ndowed with a nondegenerate metric g i n this paper we study the class of z2 gradings of c v g which are somehow compatible with the multivector structure of the grassmann algebra over v a complete characterization for such z2 gradings is obtained by classifying all the even subalgebras coming from them an expression relating such subalgebras to the usual even part of c v g is also obtained finally we employ this framework to define spinor spaces and to parametrize all the possible signature changes on c v g by z2 gradings of this algebra

a family of centered forms for rational functions abstract a family of centered forms for rational functions dependent on a parameter is investigated particular choices of the parameter are investigated in further detail it is also shown that a centered form of minimum width can be chosen from the family these centered forms are then generalized to a family of forms of higher order

non abelian chern simons action is topological invariant on 3 simple knot under su 2 gauge transformation the non abelian chern simons action is invariant on a class of three dimensional manifold 3 simple knot

an example of absence of turbulence for any reynolds number ii we study a viscous incompressible fluid moving in a two dimensional flat torus 0 l 0 2 l 2 we show a set of external forces for which the stationary state is attractive for any reynolds numberr moreover the size of this set and the basin of attraction are independent ofr

fixed point limit behavior of n mode truncated navier stokes equations as n increases the fixed point behavior ofn mode truncations of the navier stokes equations on a two dimensional torus is investigated asn increases fromn 44 on the behavior does not undergo any qualitative change furthermore the bifurcations occur at critical parameter values which clearly tend to stabilize asn approaches 100

regular and chaotic motion of fluid particles in a two dimensional fluid the authors study the chaotic behaviour exhibited by particles which move in a two dimensional fluid the connection of this lagrangian chaos with the velocity field behaviour is discussed both in the lorenz model and in truncated navier stokes equations they indicate a possible method for the onset of lagrangian chaos which seems to be rather generic lagrangian chaos appears when the eulerian equation passes from a steady solution to a periodic one via hopf bifurcation it is also shown that the transition to chaos for the velocity field eulerian chaos does not affect the particle motion properties in some typical cases

grassmann calculus pseudoclassical mechanics and geometric algebra a reformulation of grassmann calculus is presented in terms of geometric algebra a unified language for physics based on clifford algebra in this reformulation grassmann generators are replaced by vectors so that every product of generators has a natural geometric interpretation the calculus introduced by berezin the method of second quantization academic new york 1966 is shown to be unnecessary amounting to no more than an algebraic contraction this approach is not only conceptually clearer but it is also computationally more efficient as demonstrated by treatments of the grauss integral and the grassmann fourier transform the reformulation is applied to pseudoclassical mechanics ann phys 104 336 1977 where it is shown to lead to a new concept the multivector lagrangian to illustrate this idea the three dimensional fermi oscillator is reformulated and solved and its symmetry properties discussed as a result a new and highly compact formula for generating super lie algebras is revealed the paper ends with a discussion of quantization outlining a new approach to fermionic path integrals

least squares fitting of two 3 d point sets two point sets pi and p i i 1 2 n are related by p i rpi t ni where r is a rotation matrix t a translation vector and ni a noise vector given pi and p i we present an algorithm for finding the least squares solution of r and t which is based on the singular value decomposition svd of a 3 3 matrix this new algorithm is compared to two earlier algorithms with respect to computer time requirements

truncated navier stokes equations continuous transition from a five mode to a seven mode model a two parameter family of nonlinear differential equations x f x r is studied which allows one to connect continuously as varies from zero to one the different phenomenologies exhibited by a model of 5 mode truncated navier stokes equations and by a 7 mode one extending it a critical value is found for at which the most significant phenomena of the 5 mode system either vanish or go to infinity new phenomena arise then leading to the 7 mode model

conformal transformations and the effective action in the presence of boundaries the conformal properties of the heat kernel expansion are used to determine the local form of the coefficients in a manifold with boundary the conformal transformation of the effective action is obtained a novel derivation of the boundary term in the gauss bonnet chern theorem is detailed

grobner bases for finite temperature quantum computing and their complexity following the recent approach of using order domains to construct grobner bases from general projective varieties we examine the parity and time reversal arguments relating to the wightman axioms of quantum field theory and propose that the definition of associativity in these axioms should be introduced a posteriori to the cluster property in order to generalize the anyon conjecture for quantum computing to indefinite metrics we then show that this modification which we define via ideal quotients does not admit a faithful representation of the braid group because the generalized twisted inner automorphisms that we use to reintroduce associativity are only parity invariant for the prime spectra of the exterior algebra we then use a coordinate prescription for the quantum deformations of toric varieties to show how a faithful representation of the braid group can be reconstructed and argue that for a degree reverse lexicographic monomial ordered grobner basis the complexity class of this problem is bounded quantum polynomial

methods for bounding the range of a polynomial a class of methods for computing including estimates for the range of a polynomial over an interval is defined the two special cases of the bernstein form and the b spline form are shown to be members of the class

lie groups as spin groups it is shown that every lie algebra can be represented as a bivector algebra hence every lie group can be represented as a spin group thus the computational power of geometric algebra is available to simplify the analysis and applications of lie groups and lie algebras the spin version of the general linear group is thoroughly analyzed and an invariant method for constructing real spin representations of other classical groups is developed moreover it is demonstrated that every linear transformation can be represented as a monomial of vectors in geometric algebra

asymptotic analysis of the navier stokes equations abstract new bounds are established on the number of modes which determine the solutions of the navier stokes equations in two dimensions the best bound available at present is nearly proportional to the generalized grashof number defined in the paper and less than logarithmically dependent on the spatial structure or the shape of the force driving the flow to the extent than for the case of 2 dimensional rayleigh benard convection the generalized grashof number may be identified with the usual grashof number the resulting bound on the number of modes is found to differ only slightly from a bound obtained earlier on heuristic grounds

local quasi interpolants based on special multivariate quadratic spline space over a refined quadrangulation in this paper we describe the construction of a suitable normalized b spline representation for special multivariate quadratic spline space s 2 1 0 d over a refined quadrangulation we then develop as an application a general theory of quasi interpolants based on this representation

rational circular complex centered forms abstract a discussion is given of the problem of computing including estimates of the range of a complex rational function over a circular complex interval for this purpose rational circular complex centered forms are defined explicit formulas are given for the first few forms and these formulas are used to prove that forms of higher order are an improvement over the forms of lower order the forms are furthermore shown to be quadratically convergent a semi centered form is also discussed this form is shown to be quadratically convergent depending on some conditions on the coefficients of the polynomials defining the complex rational function finally a number of numerical examples are given showing the improvements obtained using the circular centered forms as compared to simple circular complex rational function estimations