g2 cubic transition between two circles with shape control this paper describes a method for joining two circles with an s shaped or with a broken back c shaped transition curve composed of at most two spiral segments in highway and railway route design or car like robot path planning it is often desirable to have such a transition it is shown that a single cubic curve can be used for blending or for a transition curve preserving g 2 continuity with local shape control parameter and more flexible constraints provision of the shape parameter and flexibility provide freedom to modify the shape in a stable manner which is an advantage over previous work by meek walton sakai and habib

a theory of algebraic cocycles we introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an eilenberg maclane space using these cocycles we develop a cohomology theory for complex algebraic varieties the theory is bigraded functorial and admits gysin maps it carries a natural cup product and a pairing to l homology chern classes of algebraic bundles are defined in the theory there is a natural transformation to singular integral cohomology theory that preserves cup products computations in special cases are carried out on a smooth variety it is proved that there are algebraic cocycles in each algebraic rational p p cohomology class in this announcement we present the outlines of a cohomology theory for algebraic varieties based on a new concept of an algebraic cocycle details will appear in fl our cohomology is a companion to the l homology theory recently studied in l f l f1 l f2 fm this homology is a bigraded theory based directly on the structure of the space of algebraic cycles it admits a natural transformation to integral homology that generalizes the usual map taking a cycle to its homology class our new cohomology theory is similarly bigraded and based on the structure of the space of algebraic cocycles it carries a ring structure coming from the complex join an elementary construction of projective geometry and it admits a natural transformation to integral cohomology chern classes are defined in the theory and transform under to the usual ones our definition of cohomology is very far from a duality construction on l homology nonetheless there is a natural and geometrically defined kronecker pairing between our morphic cohomology and l homology the foundation stone of our theory is the notion of an effective algebraic cocycle which is of some independent interest roughly speaking such a cocycle on a variety x with values in a projective variety y is a morphism from x to the space of cycles on y when x is normal this is equivalent by graphing to a cycle on x y with equidimensional fibres over x such cocycles abound in algebraic geometry and arise naturally in many circumstances the simplest perhaps is that of a flat morphism f x y whose corresponding cocycle associates to x x the pullback cycle f x many more arise naturally from synthetic constructions in projective geometry we show that every variety is rich in cocycles indeed if 1991 mathematics subject classification primary 14f99 14c05

triangulations of subanalytic sets and locally subanalytic manifolds if two polyhedrons are locally subanalytically homeomorphic that is the graph is locally subanalytic they are pl homeomorphic a lo cally subanalytic manifold is one whose coordinate transformations are locally subanalytic it is proved that a locally subanalytic manifold has a unique pl manifold structure a semialgebraic manifold also is considered

grothendieck standard conjectures morphic cohomology and hodge index theorem using morphic cohomology we produce a sequence of conjectures called morphic conjectures which terminates at the grothendieck standard conjecture a a refinement of hodge structures is given and with the assumption of morphic conjectures we prove a hodge index theorem we answer a question of friedlander and lawson by assuming the grothendieck standard conjecture b and prove that the topological filtration from morphic cohomology is equal to the grothendieck arithmetic filtration for some cases

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

algebraic cycles and infinite loop spaces in this paper we use recent results about the topology of chow varieties to answer an open question in infinite loop space theory that is we construct an infinite loop space structure on a certain product of eilenberg maclane spaces so that the total chern map is an infinite loop map an analogous result for the total stiefel whitney map is also proved further results on the structure of stabilized spaces of alebraic cycles are obtained and computational consequences are also outlined

approximation of a planar cubic bezier spiral by circular arcs a planar cubic bezier curve that is a spiral i e its curvature varies monotonically does not have internal cusps loops and inflection points it is suitable as a design tool for applications in which fair curves are important since it is polynomial it can be conveniently incorporated in cad systems that are based on b splines bezier curves or nurbs when machining objects it is desirable that as much as possible of a curved toolpath be approximated by a sequence of circular arcs rather than straight line segments such an arc spline approximation of a planar cubic bezier spiral is presented

random traveling wave and bifurcations of asymptotic behaviors in the stochastic kpp equation driven by dual noises abstract in this paper we study the influences of dually environmental noises on the traveling wave which develops from the deterministic kpp equation we prove that if the strengths of noises satisfy some condition the solution of the stochastic kpp equation with heaviside initial condition develops a random traveling wave whose wave speed is deterministic and depends on the strengths of noises if the strengths of noises satisfy some other conditions the solution tends to zero as time tends to infinity therefore there exist bifurcations of asymptotic behaviors of solution induced by the strengths of dual noises

decomposing trimmed surfaces using the voronoi diagram and a scan line algorithm many applications deal with the rendering of trimmed surfaces and the generation of grids for trimmed surfaces usually a structured or unstructured grid must be constructed in the parameter space of the trimmed surface trimmed surfaces not only cause problems in the context of grid generation but also when exchanging data between different cad systems this paper describes a new approach for decomposing the valid part of the parameter space of a trimmed surface into a set of four sided surfaces the boundaries of these four sided surfaces are line segments segments of the trimming curves themselves and segments of bisecting curves that are defined by a generalized voronoi diagram implied by the trimming curves in parameter space we use a triangular background mesh for the computation of the bisecting curves of the generalized voronoi diagram

effectiveness non effectiveness in semialgebraic and pl geometry f acquistapace r benedetti and f broglia dipartimento di matematica universit i di pisa via f buonarroti 2 1 56127 pisa italy oblatum 14 iv 1989 25 i 1990 0 introduction and statement of the main results working in real semialgebraic geometry one could easily have the feeling that the main facts can be realized by algorithms e g path connecting points triangul ations good stratifications and the main trouble is to construct reasonably fast algorithms in fact it is this in principle constructive nature of semialgebraic geometry and the existence of powerful computers which make it interesting in view of concrete applications on the other hand nabutovsky s examples of non recursive functions associ ated to natural semialgebraic constructions see i n tell us that there are some conceptual limitations to the above attitude in this paper we consider some further examples of effectiveness non effect iveness results in semialgebraic and pl geometry our starting point was the study of shiota yokoi solution of the subanalytic hence semialgebraic hauptvermutung sy and the attempt to make it effective before starting we need to introduce some notations we shall use systematically the notation

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

techniques computations and conjectures for semi topological k theory we establish the existence of an atiyah hirzebruch like spectral sequence relating the morphic cohomology groups of a smooth quasi projective complex variety to its semi topological k groups this spectral sequence is compatible with and indeed is built from the motivic spectral sequence that relates the motivic cohomology and algebraic k theory of varieties and it is also compatible with the classical atiyah hirzebruch spectral sequence in algebraic topology in the second part of this paper we use this spectral sequence in conjunction with another computational tool that we introduce namely a variation on the integral weight filtration of the borel moore singular homology of complex varieties introduced by h gillet and c soule to compute the semi topological k theory of a large class of varieties in particular we prove that for curves surfaces toric varieties projective rational three folds and related varieties the semi topological k groups and topological k groups are isomorphic in all degrees permitted by cohomological considerations we also formulate integral conjectures relating semi topological k theory to topological k theory analogous to more familiar conjectures namely the quillen lichtenbaum and beilinson lichtenbaum conjectures concerning mod n algebraic k theory and motivic cohomology in particular we prove a local vanishing result for morphic cohomology which enables us to formulate precisely a conjectural identification of morphic cohomology by a suslin our computations verify that these conjectures hold for the list of varieties above

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

stochastic generalised kpp equations we classify multiplicative white noise perturbations k dw of generalised kpp equations and their effects on deterministic approximate travelling wave solutions by the behaviour of the solutions of the stochastic generalised kpp equations converge to deterministic approximate travelling waves and if being an associated potential energy s a solution of the corresponding classical mechanical equations of newton d being a certain domain in r1 rr then the white noise perturbations essentially destroy the wave structure and force the solutions to die down for the case suppose the existence of the limit we show that there is a residual wave form but propagating at a different speed from that of the unperturbed equations numerical solutions are included and give good agreement with theoretical results

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

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

monopoles non linear σ models and two fold loop spaces in this paper we study the topology of hat m k the moduli spaces ofsu 2 monopoles associated with the yang mills higgs and bogomol nyi equations and m k non linear models from quantum field theory beautiful work of donaldson 18 19 hitchin 24 25 and taubes 37 39 40 shows that gauge equivalence classes of monopoles correspond to based rational self maps of the riemann sphere similarly the non linear models we consider here are based harmonic maps from the riemann sphere to complex projectivem space in seminal work segal 35 studied m k the space of based rational maps from the riemann sphere to complex projectivem space of a fixed degreek any element of m k is clearly an element of k2cp m the space of all based continuous maps from the riemann sphere to complex projectivem space of a fixed degreek and this assignment gives rise to the natural inclusion of m k in k2cp m segal showed that these natural inclusions are homotopy equivalences through dimensionk 2m 1 as the topology of the two fold loop space 2cp m is well understood segal s result gives a very efficient way to explicitly determine the low dimensional topology of m k thus iterated loop spaces have much to say about the topology of monopoles and non linear models

algebraic cycles and homotopy theory examine its homotopy groups which after a certain idealization of the space turn out to be astonishingly simple in fact for complex projective space pn the structure can be understood completely this yields new information about the topology of the classical chow varieties and establishes an explicit relationship with universal cohomology operations in the general case complete computations are difficult but we shall establish a complex suspension theorem and lay the foundations for a theory based on the homotopy groups of chow varieties to state the results we must give precise meaning to the space of subvarieties we begin with the fundamental case of subvarieties in complex projective n space pln for each pair of integers p and d with d 1 and 0 p n consider the set wp d pn of all finite formal sums

resultants and the algebraicity of the join pairing on chow varieties the chow van der waerden approach to algebraic cycles via resultants is used to give a purely algebraic proof for the algebraicity of the complex suspension the algebraicity of the join pairing on chow varieties then follows the approach implies a more algebraic proof of lawson s complex suspension theorem in characteristic 0 the continuity of the action of the linear isometries operad on the group completion of the stable chow variety is a consequence

a further generalisation of the planar cubic bezier spiral spiral segments are useful in the design of fair curves they are important in cad cam applications the design of highway and railway routes trajectories of mobile robots and other similar applications 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 generalises earlier results on planar cubic bezier spiral segments and examines techniques for curve design using the new results

o minimal hauptvermutung for polyhedra i milnor discovered two compact polyhedra which are homeomorphic but not pl homeomorphic a counterexample to the hauptvermutung he constructed the homeomorphism by a finite procedure repeated infinitely often informally we call a procedure constructive if it consists of an explicit procedure that is repeated only finitely many times in this sense milnor did not give a constructive procedure to define the homeomorphism between the two polyhedra in the case where the homeomorphism is semialgebraic the author and yokoi proved that the polyhedra in r n are pl homeomorphic in that article the required pl homeomorphism was not constructively defined from the given homeomorphism in the present paper we obtain the pl homeomorphism by a constructive procedure starting from the homeomorphism we prove in fact that for any ordered field r equipped with any o minimal structure two definably homeomorphic compact polyhedra in r n are pl homeomorphic the o minimal hauptvermutung theorem 1 1 together with the fact that any compact definable set is definably homeomorphic to a compact polyhedron we can say that o minimal topology is tame

intersections via resolutions abstract we investigate the viability of defining an intersection product on algebraic cycles on a singular algebraic variety by pushing forward intersection products formed on a resolution of singularities for varieties with resolutions having a certain structure including all varieties over a field of characteristic zero we obtain a stratification which reflects the geometry of the centers and the exceptional divisors this stratification is sufficiently fine that divisors can be intersected with r cycles for r 1 and 2 cycles can be intersected on a fourfold provided their incidences with the strata are controlled similar pairings are defined on a variety with one dimensional singular locus

algebraic cycles on real varieties and z 2 equivariant homotopy theory in this paper the spaces of algebraic cycles on a real projective variety x are studied as z 2 spaces under the action of the galois group gal c r in particular the equivariant homotopy type of the group of algebraic p cycles zp pc is computed a version of lawson homology for real varieties is proposed the real lawson homology groups are computed for a class of real varieties

the stable manifold theorem for stochastic differential equations type equations are treated starting with the existence of a stochastic flow for a sde we introduce the notion of a hyperbolic stationary trajectory we prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution for stratonovich sdes the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde the proof of the stable manifold theorem is based on ruelle oseledec multiplicative ergodic theory

some remarks concerning mod nk theory the spectral sequence predicted by a beilinson relating motivic cohomology to algebraic k theory has been established for smooth quasi projective varieties over a field cf fs l1 among other properties verified this spectral sequence has the expected multiplicative behavior involving cup product in motivic cohomology and product in algebraic k theory and a good multiplicative mod n reduction relating mod n motivic cohomology to mod n algebraic k theory the purpose of this short paper is to draw two simple conclusions concerning mod n algebraick theory of smooth quasi projective varieties over an algebraically closed field by comparing this spectral sequence to its localization which we show converges to mod n etale k theory in theorem 2 1 we verify that the map from mod n algebraic k theory to mod n etale k theory of a smooth quasi projective variety x over an algebraically closed field in which n is invertible is surjective in degrees greater than or equal to twice the dimension of x this refines the surjectivity result of dfst and extends an argument by a suslin in s1 then in example 3 4 we show that certain projective smooth 3 folds x constructed by s bloch and h esnault have non zero elements in k0 x z n which are killed by multiplication by a sufficiently high power of the bott element k2 x z n although the existence of such examples is perhaps not surprising there have been attempts to prove that multiplication by is always injective the authors gratefully acknowledge useful conversations with h esnault and a suslin

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

on travelling waves for the stochastic fisher kolmogorov petrovsky piscunov equation this paper is concerned with properties of the wave speed for the stochastically perturbed fisher kolmogorov petrovsky piscunov fkpp equation it was shown in the classical 1937 paper by kolmogorov petrovsky and piscunov that the large time behavior of the solution to the fkpp equation with heaviside initial data is a travelling wave in a seminal 1995 paper mueller and sowers proved that this also holds for a stochastically perturbed fkpp equation the wave speed depends on the strength of the noise in this paper bounds on the asymptotic behavior of the wave speed c as 0 and are obtained

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

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

interacting particles the stochastic fisher kolmogorov petrovsky piscounov equation and duality abstract the stochastic fisher kolmogorov petrovsky piscunov equation is t u x t d xx u u 1 u e u 1 u x t for 0 u 1 where x t is a gaussian white noise process in space and time here d and e are parameters and the equation is interpreted as the continuum limit of a spatially discretized set of ito equations solutions of this stochastic partial differential equation have an exact connection to the a a a reaction diffusion system at appropriate values of the rate coefficients and particles diffusion constant this relationship is called duality by the probabilists it is not via some hydrodynamic description of the interacting particle system in this paper we present a complete derivation of the duality relationship and use it to deduce some properties of solutions to the stochastic fisher kolmogorov petrovsky piscunov equation

harnack thom theorem for higher cycle groups and picard varieties we generalize the harnack thom theorem to relate the ranks of the lawson homology groups with 22 coefficients of a real quasiprojective variety with the ranks of its reduced real lawson homology groups in the case of zero cycle group we recover the classical harnack thom theorem and generalize the classical version to include real quasiprojective varieties we use weil s construction of picard varieties to construct reduced real picard groups and milnor s construction of universal bundles to construct some weak models of classifying spaces of some cycle groups these weak models are used to produce long exact sequences of homotopy groups which are the main tool in computing the homotopy groups of some cycle groups of divisors we obtain some congruences involving the picard number of a nonsingular real projective variety and the rank of its reduced real lawson homology groups of divisors

algebraic cycles representing cohomology operations in this paper we will show that certain universal homology classes which are fundamental in topology are algebraic to be specific the products of eilenberg maclane spaces mathcal k 2q equiv k mathbf z 2 times k mathbf z 4 times cdots times k mathbf z 2q have models which are limits of complex projective varieties precisely we have mathcal k 2q varinjlim nolimits d rightarrow infty mathcal c q d mathbf p n where mathcal c q d mathbf p n denotes the chow variety of effective cycles of codimension q and degree d on mathbf p mathbf c n it is natural to ask which elements in the homology of mathcal k 2q are represented by algebraic cycles in these approximations in this paper we find such representations for the even dimensional classes which are known as steenrod squares as well as their pontrjagin and join products these classes are dual to the cohomology classes which correspond to the basic cohomology operations also known as the steenrod squares

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

vanishing theorems for real algebraic cycles we establish the analogue of the friedlander mazur conjecture for teh s reduced lawson homology groups of real varieties which says that the reduced lawson homology of a real quasi projective variety x vanishes in homological degrees larger than the dimension of x in all weights as an application we obtain a vanishing of homotopy groups of the mod 2 topological groups of averaged cycles and a characterization in a range of indices of dos santos real lawson homology as the homotopy groups of the topological group of averaged cycles we also establish an equivariant poincare duality between equivariant friedlander walker real morphic cohomology and dos santos real lawson homology we use this together with an equivariant extension of the mod 2 beilinson lichtenbaum conjecture to compute some real lawson homology groups in terms of bredon cohomology

completions and fibrations for topological monoids we show that for a certain class of topological monoids there is a homotopy equivalence between the homotopy theoretic group completion m of a monoid m in that class and the topologized grothendieck group m associated to m the class under study is broad enough to include the chow monoids effective cycles associated to a projective algebraic variety and also the infinite symmetric products of finite cw complexes we associate principal fibrations to the completions of pairs of monoids showing the existence of long exact sequences for the naive approach to lawson homology fri91 lf91 a another proof of the eilenberg steenrod axioms for the functors x sp x in the category of finite cw complexes dold thom theorem dt56 is obtained

lawson homology for varieties with small chow groups and the induced filtration on the griffiths groups these groups form an intriguing set of invariants despite the vast literature on this subject the groups have been calculated only in very few cases such as for the projective spaces one of the goals of this note is to understand these invariants in case the variety has few cycles in the sense that rational and homological equivalence coincide for all m cycles up to a certain rank m s actually the results only deal with that part of lawson homology that is not visible in ordinary homology in that it is in the kernel of the cycle class maps cm k 2m lmhk 2m x hk 2m x

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

geometry processing for design and manufacturing this text includes papers covering topics in geometry processing applications such as surface surface intersections and offset surfaces present methods fundamental to geometric modelling are highlighted

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

geometric constraints on quadratic bezier curves using minimal length and energy this paper derives expressions for the arc length and the bending energy of quadratic bezier curves the formulas are in terms of the control point coordinates for fixed start and end points of the bezier curve the locus of the middle control point is analyzed for curves of fixed arc length or bending energy in the case of arc length this locus is convex for bending energy it is not given a line or a circle and fixed end points the locus of the middle control point is determined for those curves that are tangent to a given line or circle for line tangency this locus is a parallel line in the case of the circle the locus can be classified into one of six major types in some of these cases the locus contains circular arcs these results are then used to implement fast algorithms that construct quadratic bezier curves tangent to a given line or circle with given end points that minimize bending energy or arc length

asymptotic properties of the solutions to stochastic kpp equations a reduction method is used to prove existence and uniqueness of strong solutions to stochastic kpp equations where the initial condition may be anticipating the asymptotic behavior of the solution for large time and space and the random travelling waves are then studied under two different basic assumptions

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

quaternionic algebraic cycles and reality in this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real brauer severi varieties under the action of the galois group gal c r appropriate stabilizations of these spaces yield two equivariant spectra the first one classifies dupont seymour s quaternionic k theory and the other one classifies an equivariant cohomology theory 3 which is a natural recipient of characteristic classes kh x 3 x for quaternionic bundles over real spaces x

voronoi diagram and medial axis algorithm for planar domains with curved boundaries ii detailed algorithm description abstract details of algorithms to construct the voronoi diagrams and medial axes of planars domain bounded by free form polynomial or rational curve segments are presented based on theoretical foundations given in the first installment ramamurthy and farouki j comput appl math 1999 102 119 141 of this two part paper in particular we focus on key topological and computational issues that arise in these constructions the topological issues include i the data structures needed to represent various geometrical entities bisectors voronoi regions etc and ii the boolean operations i e union intersection and difference on planar sets required by the algorithm specifically representations for the voronoi polygons of boundary segments and for individual voronoi diagram or medial axis edges are proposed since these edges may be segments of a nonrational algebraic curves curve curve bisectors b rational curves point curve bisectors or c straight lines point point bisectors data structures tailored to each of these geometrical entities are introduced the computational issues addressed include the curve intersection algorithms required in the boolean operations and iterative schemes used to precisely locate bifurcation or n prong points n 3 of the voronoi diagram and medial axis a selection of computed voronoi diagram and medial axis examples is included to illustrate the capabilities of the algorithm

moving algebraic cycles of bounded degree the chow moving lemma is a theorem which asserts that a given algebraic s cycle on a smooth algebraic variety x can be moved within its rational equivalence class to intersect properly a given r cycle on x provided that r s dim x cf chow s2 in the past few years there has been considerable interest in studying spaces of algebraic cycles rather than simply cycles modulo an equivalence relation with this in mind it is natural to ask whether one can move a given bounded family of s cycles on the smooth variety x to intersect properly a given bounded family of r cycles the main point of this paper is to formulate and prove just such a result in theorem 3 1 we demonstrate that for any integer e and any smooth projective variety x one can simultaniously and algebraically move all effective s cycles of degree e on x so that each such cycle meets every effective r cycle of degree e on x in proper dimension the primary motivation for this moving lemma for cycles of bounded degree was the possibility of a duality theorem between cohomology and homology theories defined in terms of homotopy groups of cycle spaces using theorem 3 1 we have proved such a duality theorem for complex quasi projective varieties in f l2 we prove our moving lemma for varieties over an arbitary infinite field permitting a proof in f v of a duality theorem for motivic cohomology and homology the reader will find that our moving lemma has numerous good properties first of all the move is given as an algebraic move parametrized by a punctured projective line on chow varietes although this move is good only for s cycles of bounded degree it is defined on all effective s cycles moreover the move starts at time 0 by expressing an effective s cycle z as a difference of effective s cycles both of which have intersection properties no worse than z finally our moving lemma is applicable to smooth quasiprojective varieties for it is stated for a possibly singular projective variety x resulting in a conclusion of proper intersection off the singular locus of x the classical motivation for the moving lemma was to define an intersection product on algebraic cycles modulo rational equivalence thereby establishing the chow ring a x some of the classical literature overlooked the question of whether or not intersection of cycles defined via a moving lemma is independent of the move e g chow r s2 on the other hand cf chev s3 one direct consequence of our moving lemma is a proof for smooth quasi projective varieties that the intersection product is indeed well defined independent of the choice of move theorem 3 4 of course the intersection product now has an intrinsic formulation for all smooth algebraic varieties due to fulton

on the nonlinear diffusion equation of kolmogorov petrovsky and piscounov abstract recently bramson proved a theorem that classifies the initial data under which solutions of the k p p equation converge to the appropriate travelling waves in this paper a simplified proof is given by using maximal principles instead of his brownian motion approach the regularity condition on the forcing term is also weakened

high quality point sampling for b spline fitting of parametric curves with feature recognition abstract determining a sequence of points from parametric curves is essential for the construction of least squares b spline fitting curves in geometric modeling and related applications uniform sampling methods always determine points uniformly in arc length curvature or a weighted form but feature points intuitively indicating the curve profile are generally ignored in this paper we focus on high quality point sampling so as to better capture the original curve shape firstly all the feature points inflection points and extreme curvature points on the curve are recognized by a parabolic interpolation method then more auxiliary points are adaptively added according to the characteristic function defined as a weighted sum of arc length and total curvature characteristics by adjusting the weight in the characteristic function it provides more flexibility and controllability to obtain auxiliary points finally we convert the original curve into a b spline curve by interpolating these feature points and auxiliary points using the progressive iterative approximation method numerical examples demonstrate the effectiveness and high quality of our proposed method

graph mappings and poincare duality we give a new geometric description for a compact oriented pseduo manifold x of the poincare duality map from the integral cohomology of x to the integral homology of x our construction takes a multi valued lipschitz map on x with values in a sphere sn to its geometric measure theoretic graph in x sn and then to the slice of this graph as an integral cycle on x this construction is compatible with analogous constructions on algebraic cocycles on projective varieties employed by the authors and others

random travelling waves for the kpp equation with noise consider the stochastic partial differential equation formula where t x is two parameter while noise assume that u0 is a continuous function taking values in 0 1 such that for some constant a 0 we have c1 u0 x 1 for x a let the wavefront b t sup x r u t x 0 we show that for small enough and with probability 1 limt b t t exists and lies in 0 this limit depends only on the law of v t x u t b t x tends toward a stationary limit as t we also analyze the length of the region a t b t which is the smallest closed interval containing the points x at which 0 u t x 1 we show that the length of this region tends toward a stationary distribution thus the wavefront does not degenerate

rational isomorphisms between k theories and cohomology theories the well known isomorphism relating the rational algebraic k theory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map the segre map of infinite loop spaces moreover the associated chern character map on rational homotopy groups is shown to be a ring isomorphism a technique is introduced that establishes a useful general criterion for a natural transformation of functors on quasi projective complex varieties to induce a homotopy equivalence of semi topological singular complexes since semi topological k theory and morphic cohomology can be formulated as the semi topological singular complexes associated to algebraic k theory and motivic cohomology this criterion provides a rational isomorphism between the semi topological k theory groups and the morphic cohomology groups of a smooth complex variety consequences include a riemann roch theorem for the chern character on semi topological k theory and an interpretation of the topological filtration on singular cohomology groups in k theoretic terms

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

stiefel whitney classes and the conormal cycle of a singular variety a geometric construction of sullivan s stiefel whitney homology classes of a real analytic variety x is given by means of the conormal cycle of an embedding of x in a smooth variety we prove that the stiefel whitney classes define additive natural transformations from certain constructible functions to homology we also show that for a complex analytic variety these classes are the mod 2 reductions of the chern macpherson classes we present a new definition of the stiefel whitney homology classes of a possibly singular real analytic variety x the original definition due to sullivan s involves a triangulation of x its geometric meaning is unclear our definition uses the conormal cycle of an embedding of x in a smooth variety the conormal cycle of a subanalytic subset x of an analytic manifold m was defined by the first author f4 using geometric measure theory the conormal cycle is an integral current representing up to sign kashiwara s characteristic cycle of the sheaf dmrx f4 4 7 our construction of the stiefel whitney classes is based on the fundamental observation that the conormal cycle of a real analytic variety is antipodally symmetric mod 2 we use this observation to give a definition of stiefel whitney classes which is parallel to the first author s definition of the chern schwartz macpherson classes of a complex analytic variety f5 this definition of chern classes is related to earlier work of brylinski dubson kashiwara bdk and sabbah sa in fact we show that the stiefel whitney homology classes of a complex analytic variety are the mod 2 reductions of the chern classes we prove that our stiefel whitney classes satisfy axioms similar to the deligne grothendieck axioms for chern classes and we prove a specialization formula for the stiefel whitney classes of a family of varieties we show that the stiefel whitney classes of an affine real analytic variety x are represented by the polar cycles of x introduced for simplicial spaces by banchoff b and mccrory mc and we give a new proof of the combinatorial formula for stiefel whitney classes of manifolds our central result is a specialization formula for the conormal cycle theorem 3 7 we use this formula to prove the basic results of kashiwara and schapira s calculus of subanalytically constructible functions sc ks 9 7 as well as the pushforward and specialization formulas for stiefel whitney classes received by the editors october 2 1995 1991 mathematics subject classification primary 14p25 57r20 secondary 14p15 49q15

generalized abel jacobi map on lawson homology we construct an abel jacobi type map on the homologically trivial part of lawson homology groups it generalizes the abel jacobi map constructed by griffiths by using a result of h clemens we answer affirmatively the question whether there exists a smooth projective complex variety with infinitely generated lawson homology groups l ph 2p k x bbb q when k0 as a corollary we find for any nonnegative integer j a smooth complex projective variety x carrying infinitely generated semi topological k groups k sst j x bbb q

chow flag variety and lawson homology the chow flag varieties are introduced and studied in a family of pure dimensional pro jective complex varieties defined via chow flag varieties the isomorphism type of lawson homology of a general member in the family is discussed applications to morphic cohomology and motivic cohomology on smooth projective complex varieties are discussed 1 introduction by variety we mean a quasi projective variety defined as a zariski open subset of the zero locus of a family of homogeneous polynomials in some projective space in particular varieties may not be irreducible if x is a projective complex variety the lawson homology of x denoted as lrhi x where r and i are appropriate nonnegative integers was first considered by lawson in l and then systematically formulated and studied by friedlander in f91 properties of lawson homology have been studied and relations to other important theories have been understood however it is hard to compute lawson homology groups for a general complex variety friedlander and mazur conjectured in fm that lrhi x 0w herex is smooth projective and i is greater than twice the dimension of x this conjecture has been verified in a few situations in this paper we consider the isomorphism type of lawson homology groups of members in a family of projective complex varieties here is what we mean by a family of complex varieties let p n be the usual complex projective space and let cr d p n denote the chow variety of effective r cycles of degree d in p n for any point x in cr d p n l etz x denote the corresponding cycle a family of projective varieties defined in p n of pure dimension r is by definition a subvariety y of cr d p n a member of this family is the support of a cycle z y for some point y y the connection between the notion of families of varieties used in this paper and the usual notion of families of varieties defined via morphisms is explained in f91 theorem 14 according to f91 if y is smooth then a family of projective varieties in p n indexed by y corresponds to a morphism f v y such that the cycle associated to the scheme theoretical fiber over y y is the cycle z y a nd thatv is a subvariety of p n y and f is the restriction of the projection map p n y y if y is a family of projective varieties it is desirable to show certain statements are generically true this means there exists a closed proper subvariety z of y such that the statement is true for any y y z our main result in this paper see

a variational principle for kpp front speeds in temporally random shear flows we establish the variational principle of kolmogorov petrovsky piskunov kpp front speeds in temporally random shear flows with sufficiently decaying correlations a key quantity in the variational principle is the almost sure lyapunov exponent of a heat operator with random potential to prove the variational principle we use the comparison principle of solutions the path integral representation of solutions and large deviation estimates of the associated stochastic flows the variational principle then allows us to analytically bound the front speeds the speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length and the sublinear growth law if the temporal decorrelation is also large enough the so called bending phenomenon