on the haar measure of the quantumsu n group we prove that the haar state associated to the compact matrix quantum groupsu n is faithful for 1 1 0 and anyn 2

strong connections and chern connes pairing in the hopf galois theory we reformulate the concept of connection on a hopf galois extension b p in order to apply it in computing the chern connes pairing between the cyclic cohomology hc 2 n b and k 0 b this reformulation allows us to show that a hopf galois extension admitting a strong connection is projective and left faithfully flat it also enables us to conclude that a strong connection is a cuntz quillen type bimodule connection to exemplify the theory we construct a strong connection super dirac monopole to find out the chern connes pairing for the super line bundles associated to a super hopf fibration

quantum symmetry groups of noncommutative spheres we show that the noncommutative spheres of connes and landi are quantum homogeneous spaces for certain compact quantum groups we give a general construction of homogeneous spaces which support noncommutative spin geometries

the use of cornu spirals in drawing planar curves of controlled curvature abstract cornu spirals or clothoids have been used in highway design for many years in the past the spirals have been found manually by draftsmen the purpose of this paper is to show that it is practical to find these spirals with a micro computer the design curve will be made up of arcs of circles and segments of cornu spirals joined in such a way that the curvature is continuous throughout and takes its largest values on the arcs of circles thus the radii of the circles used will limit and control the curvature of the whole design curve

planar g 2 hermite interpolation with some fair c shaped curves g2 hermite data consists of two points two unit tangent vectors at those points and two signed curvatures at those points the planar g2 hermite interpolation problem is to find a planar curve matching planar g2 hermite data in this paper a c shaped interpolating curve made of one or two spirals is sought such a curve is considered fair because it comprises a small number of spirals the c shaped curve used here is made by joining a circular arc and a conic in a g2 manner a curve of this type that matches given g2 hermite data can be found by solving a quadratic equation the new curve is compared to the cubic bezier curve and to a curve made from a g2 join of a pair of quadratics the new curve covers a much larger range of the g2 hermite data that can be matched by a c shaped curve of one or two spirals than those curves cover

approximating smooth planar curves by arc splines when a smooth curve is used to describe the path of a computer controlled cutting machine the path is usually approximated by many straight line segments it is preferable to describe the cutting path as an arc spline a tangent continuous piecewise curve made of circular arcs and straight line segments this paper presents an algorithm for finding an arbitrarily close arc spline approximation of a smooth curve

meromorphic functions that share four values an old theorem of r nevanlinna states that if two distinct nonconstant meromorphic functions share four values counting multiplicities then the functions are mbbius transformations of each other two of the shared values are picard values for both functions and the cross ratio of a particular permutation of the shared values equals 1 in this paper we show that if two nonconstant meromorphic functions share two values counting multiplicities and share two other values ignoring multiplicities then the functions share all four values counting multiplicities

the family of biarcs that matches planar two point g1 hermite data the biarc is a curve made by joining two circular arcs in a g 1 fashion there is a one parameter family of biarcs that can match given planar two point g 1 hermite data this note considers the range of g 1 hermite data that can be matched identifies the region in which the members of the family of biarcs lie and shows that there is exactly one member biarc passing through each point in that region

the generalised cornu spiral and its application to span generation abstract a cornu spiral is a plane curve having a linear curvature profile this paper considers plane curves having rational linear curvature profiles these curves are defined to be generalised cornu spirals gcs and are quality curves in the sense that they are continuous and smooth can contain one inflection at most and have a bounded and monotonic curvature profile in addition the gcs has an extra degree of freedom over the cornu spiral that is available for shape control starting from the intrinsic equation of the gcs the technique of curve synthesis is used to design a quality curve that can be applied to a wide range of span generation problems

noncommutative instantons on the 4 sphere from quantum groups we describe an approach to the noncommutative instantons on the 4 sphere based on quantum group theory we quantize the hopf bundle 7 4 making use of the concept of quantum coisotropic subgroups the analysis of the semiclassical poisson lie structure of u 4 shows that the diagonal su 2 must be conjugated to be properly quantized the quantum coisotropic subgroup we obtain is the standard su q 2 it determines a new deformation of the 4 sphere 4 q as the algebra of coinvariants in q 7 we show that the quantum vector bundle associated to the fundamental corepresentation of su q 2 is finitely generated and projective and we compute the explicit projector we give the unitary representations of 4 q we define two 0 summable fredholm modules and we compute the chern connes pairing between the projector and their characters it comes out that even the zero class in cyclic homology is non trivial

quantum instantons with classical moduli spaces we introduce a quantum minkowski space time based on the quantum group su 2 q extended by a degree operator and formulate a quantum version of the anti self dual yang mills equation we construct solutions of the quantum equations using the classical adhm linear data and conjecture that up to gauge transformations our construction yields all the solutions we also find a deformation of penrose s twistor diagram giving a correspondence between the quantum minkowski space time and the classical projective space p 3

a generalisation of the pythagorean hodograph quintic spiral an existing planar pythagorean hodograph ph quintic spiral with zero curvature at one end is generalised to form a two parameter family of ph quintic spirals the additional degree of freedom allows specification of an ending angle of tangent and curvature its shape and curvature compares well to that of the clothoid which has applications in areas such as highway design and robot path planning it is thus a viable nonuniform rational b spline alternative for such applications it is also useful for other computer aided design or computer aided geometric design applications in which drawing curves with controlled curvature e g fair curves is important

circular spline fitting using an evolution process we propose a new method to approximate a given set of ordered data points by a spatial circular spline curve at first an initial circular spline curve is generated by biarc interpolation then an evolution process based on a least squares approximation is applied to the curve during the evolution process the circular spline curve converges dynamically to a stable shape our method does not need any tangent information during the evolution process the number of arcs is automatically adapted to the data such that the final curve contains as few arc arcs as possible we prove that the evolution process is equivalent to a gauss newton type method

planar g 2 transition with a fair pythagorean hodograph quintic curve recently planar cubic and pythagorean hodograph quintic transition curves that are suitable for g2 blending were developed they are suitable for blending e g rounding corners or for smooth transition between two curves e g two circular arcs it was shown that a single cubic segment can be used as a transition curve with the guarantee that an s shaped transition curve will have no curvature extrema and a c shaped transition curve will have a single curvature extremum the results for the cubic curve are now extended to pythagorean hodograph quintic curves a pythagorean hodograph curve has the attractive properties that its arclength is a polynomial of its parameter and its offset is rational a quintic is the lowest degree pythagorean hodograph curve that may have an inflection point pythagorean hodograph curves with no curvature extrema for an s shaped transition and a single curvature extremum for a c shaped transition are suitable for the design of fair curves e g in highway design or for blending in cad applications

meromorphic solutions of difference equations integrability and the discrete painleve equations the painlev e property is closely connected to differential equations that are integrable via related iso monodromy problems many apparently integrable discrete analogues of the painlev e equations have appeared in the literature the existence of sufficiently many finite order meromorphic solutions appears to be a good analogue of the painlev e property for discrete equations in which the independent variable is taken to be complex a general introduction to nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q difference operators new results are presented concerning equations of the form w z 1 w z 1 r z w where r is rational in w with meromorphic coefficients

planar interpolation with a pair of rational spirals spirals are curves of one signed monotone increasing or decreasing curvature spiral segments have the advantage that the minimum and maximum curvatures are at their endpoints two situations where the planar two point g 2 hermite interpolation problem can be solved with a pair of rational spiral segments are outlined here

two entire functions sharing one small function cm and two other small functions im abstract in this paper we prove a result that if two entire functions share one small function cm and two other small functions im then f is a quasi mobius transformation of g which generalizes g g gundersen s 2cm 2im theorem see g g gundersen trans am math soc 277 1983 545 567 and g g gundersen trans am math soc 304 1987 847 850 from sharing values to sharing small functions in the case where both f and g are entire functions

orthogonal grid generation for two dimensional ducts abstract we present an orthogonal trajectory method for grid generation for two dimensional ducts the method can be seen as a generalization of the polar grid for a wedge or a circular annulus transversal grid lines are formed by arcs of circles and orthogonal trajectories are found by solving initial value problems for ordinary differential equations in order to have the method working a condition involving the local width of the duct and the curvatures of the boundary curves must be satisfied this condition is easily computable and it should be checked in advance for illustration several computational examples are included

on results of h ueda and g brosch concerning the unicity of meromorphic functions by emplying the main ideas from g brosch two results of r nevanlinna s four value theorem type on the unicity of meromorphic functions in the open complex plane sharing three distinct values and a fourth value or a pair of values weakly are proved which extend and generalize previous results of r nevanlinna g g gundersen h ueda and g brosch

on modules associated to coalgebra galois extensions abstract for a given entwining structure a c involving an algebraa a coalgebrac and an entwining map c a a c a categorymca of right a c modules is defined and its structure analysed in particular the notion of a measuring of a c to a c is introduced and certain functors betweenmca andmca induced by such a measuring are defined it is shown that these functors are inverse equivalences iff they are exact or one of them faithfully exact and the measuring satisfies a certain galois type condition next left moduleseand right modules associated to ac galois extensionaofbare defined these can be thought of as objects dual to fibre bundles with coalgebracin the place of a structure group and a fibrev cross sections of such associated modules are defined as module mapse bor b it is shown that they can be identified with suitably equivariant maps from the fibre toa also it is shown that ac galois extension is cleft if and only ifa b cas leftb modules and rightc comodules the relationship between the moduleseand is studied in the case whenvis finite dimensional and in the case when the canonical entwining map is bijective

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

quantum groups on fibre bundles it is shown that the principle of locality and noncommutative geometry can be connected by a sheaf theoretical method in this framework quantum spaces are introduced and examples in mathematical physics are given within the language of quantum spaces noncommutative principal and vector bundles are defined and their properties are studied important constructions in the classical theory of principal fibre bundles like associated bundles and differential calculi are carried over to the quantum case at the endq deformed instanton models are introduced for every integral index

g3 quintic polynomial approximation for generalised cornu spiral segments within computer aided design cad there is a need to construct fair curves the generalised cornu spirals gcss are a set of curves with a monotonic curvature profile and are hence considered fair but implementation in current cad systems is not straightforward partly due to not being in the usual polynomial form a method to approximate a gcs using a quintic polynomial curve is presented the method seeks to interpolate the gcs to satisfy the g 3 constraints at the end points with a quintic bezier leaving two degrees of freedom an initial approximation is shown to be effective for the majority of gcs curves moreover it is possible to determine when an initial approximation is likely to be poor if this approximation does not meet the tolerance required a search involving two parameters is performed characteristics of the search domain are used to establish a suitable starting value

smooth polynomial approximation of spiral arcs constructing fair curve segments using parametric polynomials is difficult due to the oscillatory nature of polynomials even nurbs curves can exhibit unsatisfactory curvature profiles curve segments with monotonic curvature profiles for example spiral arcs exist but are intrinsically non polynomial in nature and thus difficult to integrate into existing cad systems a method of constructing an approximation to a generalised cornu spiral gcs arc using non rational quintic bezier curves matching end points end slopes and end curvatures is presented by defining an objective function based on the relative error between the curvature profiles of the gcs and its bezier approximation a curve segment is constructed that has a monotonic curvature profile within a specified tolerance

curvature extrema of planar parametric polynomial cubic curves parametric polynomial cubic curve segments are widely used in computer aided design and computer aided geometric design applications because their flexibility makes them suitable for use in the interactive design of curves and surfaces shape properties of these segments have been studied and results on the occurrence of cusps and whether the segment is s or c shaped are available in the literature their critical points however are not as well known this paper presents results on the number and location of curvature extrema of these segments all possible numbers of curvature extrema are found and it is demonstrated that all possible curvature extrema can be obtained numerically for any planar parametric cubic curve these results are useful in the study of the fairness of curves designed with parametric polynomial cubic curve segments

geometry of quantum principal bundles i a theory of principal bundles possessing quantum structure groups and classical base manifolds is presented structural analysis of such quantum principal bundles is performed a differential calculus is constructed combining differential forms on the base manifold with an appropriate differential calculus on the structure quantum group relations between the calculus on the group and the calculus on the bundle are investigated a concept of pseudo tensoriality is formulated the formalism of connections is developed in particular operators of horizontal projection covariant derivative and curvature are constructed and analyzed generalizations of the first structure equation and of the bianchi identity are found illustrative examples are presented

a planar cubic be zier spiral a planar cubic bezier curve segment that is a spiral i e its curvature varies monotonically with arc length is discussed since this curve segment does not have cusps loops and inflection points except for a single inflection point at its beginning it is suitable for applications such as highway design in which the clothoid has been traditionally used since it is polynomial it can be conveniently incorporated in cad systems that are based on b splines bezier curves or nurbs nonuniform rational b splines and is thus suitable for general curve design applications in which fair curves are important

non existence of rational arc length parameterizations for curves in rn we show as a generalization of prior results for r 2 and r 3 that for all n 2 the only curves in r n with rational arc length parameterizations are straight lines

an arc spline approximation to a clothoid the clothoid is a spiral used in highway and railway route design clothoids are transcendental functions and so have been approximated by polynomials by power series and continued fractions and by rational functions here the clothoid is approximated by an arc spline the chief advantage in doing so is that arc splines are very easy to lay out and to offset examples show that the approximation is of extremely high accuracy it is proved that if the arc spline has n arcs then the error in the approximation is of order o 1 n2

a note on quintic polynomial approximation of generalized cornu spiral segments in two recent papers cripps et al 2010 3 and cross and cripps 2012 2 quintic polynomial approximations of the generalized cornu spirals have been studied by considering g 2 continuity and g 3 continuity at the end points respectively the quintic curve is constructed so that the maximum curvature error is within the specified tolerance in this paper we provide corrections to the typing errors in cross and cripps 2012 2 and propose a simpler and more efficient method for the g 2 constrained quintic polynomial approximation where the g 2 conditions are always satisfied by four free variables also we introduce a new error measure of the maximum curvature error which can greatly reduce the computational time when looking for the solution numerical experiments demonstrate the effectiveness of the new measure

generalized fresnel integrals and fractal properties of related spirals abstract we obtain a new asymptotic expansion of generalized fresnel integrals x t 0 t cos q s d s for large t where q s s p when s and p 1 the terms of the expansion are defined via a simple iterative algorithm using this we show that the box dimension of the related q clothoid also called the generalized euler or cornu spiral is equal to d 2 p 2 p 1 furthermore this curve is minkowski measurable and we compute its d dimensional minkowski content we also find oscillatory dimension of fresnel integrals by studying the corresponding chirps

the best g 1 cubic and g 2 quartic bezier approximations of circular arcs we obtain cubic and quartic bezier approximations of circular arcs that respectively satisfy g 1 and g 2 end point interpolation conditions we identify the necessary and sufficient conditions for such approximations to be the best in the sense that they have the minimum hausdorff distance to the circular arc we then establish the existence and uniqueness of these best approximations and present practical methods to calculate them which are verified by examples

noncommutative riemannian and spin geometry of the standard q sphere we study the quantum sphere open image in new window as a quantum riemannian manifold in the quantum frame bundle approach we exhibit its 2 dimensional cotangent bundle as a direct sum 0 1 1 0 in a double complex we find the natural metric volume form hodge operator laplace and maxwell operators and projective module structure we show that the q monopole as spin connection induces a natural levi civita type connection and find its ricci curvature and q dirac operator open image in new window we find the possibility of an antisymmetric volume form quantum correction to the ricci curvature and lichnerowicz type formulae for open image in new window we also remark on the geometric q borel weil bott construction

approximating cornu spirals by arc splines while modeling planar parametric curves it is desirable to approximate them efficiently by arc splines with fewest segments within the prescribed tolerance planar arc splines are ideal for describing the path of a numerically controlled cutting machine because they are easy to offset in this paper we derive a vector solution to interpolate a g1 continuous biarc curve between a given pair of points with specified start joint and end tangents using this solution as the underlying idea we present an efficient algorithm for approximating a cornu spiral with a sequence of arcs within the prescribed maximum deviation the method is simple and generic and can be applied to other types of planar parametric curves as well

noncommutative manifolds the instanton algebra and isospectral deformations we give new examples of noncommutative manifolds that are less standard than the nc torus or moyal deformations of n they arise naturally from basic considerations of noncommutative differential topology and have non trivial global features

clothoid spline transition spirals highway and railway designers use clothoid splines planar g2 curves consisting of straight line segments circular arcs and clothoid segments as center lines in route location this paper considers the problem of finding a clothoid spline transition spiral which joins two given points and matches given curvatures and unit tangents at the two points conditions are given for the existence and uniqueness of the clothoid spline transition spirals and algorithms for finding them are outlined

a hopf bundle over a quantum four sphere from the symplectic group we construct a quantum version of the su 2 hopf bundle s7 s4 the quantum sphere s7q arises from the symplectic group spq 2 and a quantum 4 sphere s4q is obtained via a suitable self adjoint idempotent p whose entries generate the algebra a s4q of polynomial functions over it this projection determines a deformation of an anti instanton bundle over the classical sphere s4 we compute the fundamental k homology class of s4q and pair it with the class of p in the k theory getting the value 1 for the topological charge there is a right coaction of suq 2 on s7q such that the algebra a s7q is a non trivial quantum principal bundle over a s4q with structure quantum group a suq 2

rational approximations for the fresnel integrals a class of simple rational polynomial approximations for the fresnel integrals is given with maximum errors from 1 7 x 10 3 down to 4 x 10 8 the domain 0 oo is not subdivided the format is particularly convenient for programmable hand calculators and microcomputer subroutines the best known algorithms for computer evaluation of the fresnel integrals 1 c x is x j exp ip t2 dt 0 are those of hastings 1 41 and of boersma 2 the former is of very limited precision the latter is relatively cumbersome and is subject to nonobvious underflow subtraction errors as is direct evaluation of the standard maclaurin and asymptotic series an extensive set of rational polynomial approximations has been given by cody 3 these are efficient for very high precision work but tedious for intermediate levels of precision because the domain is subdivided into five intervals our approach follows hastings but recasts the formulas into the polar form 2 c x 2ri x sinf1pr ajk x x2 i 3 s x 2 rim x cos 7t ajk x x2 where the rim and ajk functions are rational approximations of the form i m2 4 r 01 l c1xi e dix i o i 0 i k s ajk e a x e b x o i o the domain of x is positive real numbers from zero to infinity we consider error functions of the form 6 sr rim ro 7 sa 7tro ak ao where r0 and ao are exact values these functions represent orthogonal errors in the plane of the cornu spiral with the coefficient inserted in 7 to make the magnitudes received may 2 1983 revised january 9 1984 1980 mathematics subject classification primary 65d20 secondary 33a20 41a50 78a50

quintic polynomial approximation of log aesthetic curves by curvature deviation log aesthetic curves lacs possessing monotone curvature and including many classical curves have been widely used to describe fair shapes in geometric modeling however they are generally represented in non polynomial form and are thus not compatible with current cad systems in this paper we present quintic polynomial approximation of lac segments for a given lac segment a quintic g 2 interpolating bezier curve is obtained by minimizing a curvature based error metric with the advantage of being more likely to preserve the monotone curvature property numerical experiments demonstrate that our method can usually generate better results than the previous methods in terms of the deviation in positions and curvatures

quantum spheres and projective spaces as graph algebras the c algebras of continuous functions on quantum spheres quantum real projective spaces and quantum complex projective spaces are realized as cuntz krieger algebras corresponding to suitable directed graphs structural results about these quantum spaces especially about their ideals and k theory are then derived from the general theory of graph algebras it is shown that the quantum even and odd dimensional spheres are produced by repeated application of a quantum double suspension to two points and the circle respectively

generation of log aesthetic curves using adaptive runge kutta methods log aesthetic curve lac has been explored extensively by many researchers since 2005 at first gaussian kronrod has been proposed to evaluate lac as the formulation of lac involves double integration recently incomplete gamma function igf has been proposed to represent lac analytically which decreases the computation time up to 13 times this paper embarks on the representation of lac using adaptive runge kutta methods to decrease the lac computation time the famous adaptive methods such as runge kutta fehlberg dormand prince sarafyan and kutta merson are employed to evaluate lac so that a desired accuracy can be achieved this paper ends with a detailed investigation on performance metric of igf and adaptive rk methods to compute lac these methods will be compared in terms of computation time and truncation error numerical results indicate that the computation time of lac can be greatly improved and at the same time preserving the lac s family

instantons on the quantum 4 spheres s 4_q we introduce noncommutative algebras a q of quantum 4 spheres s 4 q with q defined via a suspension of the quantum group su q 2 and a quantum instanton bundle described by a selfadjoint idempotent e mat4 a q e 2 e e contrary to what happens for the classical case or for the noncommutative instanton constructed in 8 the first chern connes class ch 1 e does not vanish thus signaling a dimension drop the second chern connes class ch 2 e does not vanish as well and the couple ch 1 e ch 2 e defines a cycle in the b b bicomplex of cyclic homology

an improvement of the nevanlinna gundersen theorem a well known result of nevanlinna states that for two nonconstant meromorphic functions f and g on the complex plane c and for four distinct values aj c if f aj g aj for all 1 j 4 then g is a mobius transformation of f in 1983 gundersen generalized the result of nevanlinna to the case where the above condition is replaced by min f aj 1 min g aj 1 for j 1 2 and f aj g aj for j 3 4 in this paper we prove that the theorem of gundersen remains valid to the case where min f aj 1 min g aj 1 for j 1 2 and min f aj 2 min g aj 2 for j 3 4 furthermore we work on the case where aj are small functions

a note on finding clothoids the clothoid is a spiral that is used as transition curve in highway and railway route design although its defining formulas are transcendental functions recent work has shown that it can be used fairly easily on small computers a single nonlinear equation must be solved in each of four common arrangements of the clothoid the purpose of this work is to give starting intervals for which newton s root finding method applied to those equations is guaranteed to converge

meromorphic functions that share four small functions in this paper we prove that if two nonconstant meromorphic functions f and g share two small functions cm and share other two small functions im then f must be a quasi mobius transformation of g this result is a generalization of several results obtained by g g gundersen and li yang

efficient robust approximation of the generalised cornu spiral a generalised cornu spiral gcs is a planar curve defined to have a monotonic rational linear curvature profile and as such these curves are considered fair however their implementation in current cad systems is not straight forward partly due to not being in the usual polynomial form a gcs cannot be expressed exactly using a finite polynomial and so a compromise can be achieved by instead approximating the gcs with a suitable polynomial an efficient robust approximation of the gcs using quintic polynomials is presented the approximation satisfies the g 2 continuity conditions at the end points and the remaining four degrees of freedom are argued for by looking at g 3 approximations the method begins by reparameterising the gcs in terms of more intuitive geometric descriptions the winding angle change in curvature and a shape factor the g 3 approximations provide insight to help define values for the free parameters and the new geometric form allows for the shortcomings in the g 3 approximations to be controlled the efficiency of the approximation is improved compared to earlier methods which required a numerical search also there is strong evidence that the method guarantees a satisfactory approximation when the gcs lies within certain identified bounds

planar g 2 transition curves composed of cubic bezier spiral segments in curve and surface design it is often desirable to have a planar transition curve composed of at most two spiral segments between two circles the purpose may be practical e g in highway design or aesthetic cubic bezier curves are commonly used in curve and surface design because they are of low degree are easily evaluated and allow inflection points this paper generalizes earlier results on planar cubic bezier spiral segments which were proposed as transition curve elements and examines techniques for curve design using the new results

projective module description of the q monopole the dirac q monopole connection is used to compute projector matrices of quantum hopf line bundles for arbitrary winding number the chern connes pairing of cyclic cohomology and k theory is computed for the winding number 1 the non triviality of this pairing is used to conclude that the quantum principal hopf fibration is non cleft among general results we provide a left right symmetric characterization of the canonical strong connections on quantum principal homogeneous spaces with an injective antipode we also provide for arbitrary strong connections on algebraic quantum principal bundles hopf galois extensions their associated covariant derivatives on projective modules

deformation quantization of heisenberg manifolds form a smooth manifold equipped with a poisson bracket we formulate ac algebra framework for deformation quantization including the possibility of invariance under a lie group of diffeomorphisms preserving the poisson bracket we then show that the much studied non commutative tori give examples of such deformation quantizations invariant under the usual action of ordinary tori going beyond this the main results of the paper provide a construction of invariant deformation quantizations for those poisson brackets on heisenberg manifolds which are invariant under the action of the heisenberg lie group and for various generalizations suggested by this class of examples interesting examples are obtained of simplec algebras on which the heisenberg group acts ergodically

quantum even spheres σ2nq from poisson double suspension we define even dimensional quantum spheres 2nq that generalize to higher dimension the standard quantum two sphere of podle and the four sphere 4q obtained in the quantization of the hopf bundle the construction relies on an iterated poisson double suspension of the standard podle two sphere the poisson spheres that we get have the same kind of symplectic foliation consisting of a degenerate point and a symplectic 2n and after quantization have the same c algebraic completion we investigate their k homology and k theory by introducing fredholm modules and projectors

g1 interpolation with a single cornu spiral segment cornu spiral segments are used in applications such as the geometric design of highways and railways robot path planning and shape completion for some applications e g shape completion in computer vision it is important to use a single visually pleasing curve segment to smoothly fill a gap even though the gap may not be filled in a curvature continuous manner an improved method for doing so using a cornu spiral segment is discussed the method is generally suitable for any application where it is required to smoothly fit a curve between two given points when the corresponding tangent directions but not the curvatures are also given or known

θ deformations as compact quantum metric spaces let m be a compact spin manifold with a smooth action of the n torus connes and landi constructed deformations m of m parameterized by n n real skew symmetric matrices the m s together with the canonical dirac operator d open image in new window on m are an isospectral deformation of m the dirac operator d defines a lipschitz seminorm on c m which defines a metric on the state space of c m we show that when m is connected this metric induces the weak topology this means that m is a compact quantum metric space in the sense of rieffel

correction to meromorphic functions that share four values trans amer math soc 277 1983 no 2 545 567 mr0694375 84g 30028

on the g2 hermite interpolation problem with clothoids abstract the g 2 hermite interpolation problem with clothoid curves requires to find the interpolating clothoid that matches initial and final positions tangents and curvatures also known as g 2 hermite data in the paper we prove that this problem does not always admit solution with only one clothoid segment nor with two as some counterexamples show the general fitting scheme herein proposed requires three arcs determined via the solution of a nonlinear system of 8 equations in 10 unknowns we discuss how it is possible to recast this system to 2 equations and how to efficiently solve it by means of the newton method the choice of the clothoid is crucial because it exhibits the curvature which is linear with the arc length an important property in many applications ranging from path planning for autonomous vehicles road design manufacturing and graphics the algorithm is tested on a fine hypercube of all possible configurations of angles and curvatures it always converges and in the worst case it requires 5 standard newton iterations