stochastic weighted particle methods for population balance equations a class of coagulation weight transfer functions is constructed each member of which leads to a stochastic particle algorithm for the numerical treatment of population balance equations these algorithms are based on systems of weighted computational particles and the weight transfer functions are constructed such that the number of computational particles does not change during coagulation events the algorithms also facilitate the simulation of physical processes that change single particles such as growth or other surface reactions four members of the algorithm family have been numerically validated by comparison to analytic solutions to simple problems numerical experiments have been performed for complex laminar premixed flame systems in which members of the class of stochastic weighted particle methods were compared to each other and to a direct simulation algorithm two of the weighted algorithms have been shown to offer performance advantages over the direct simulation algorithm in situations where interest is focused on the larger particles in a system the extent of this advantage depends on the particular system and on the quantities of interest

real algebraic hypersurfaces in complex projective varieties let m be a compact affine nonsingular real algebraic variety of dimension m for the background material on real algebraic geometry the reader may refer to the book 2 by an affine real algebraic variety we mean a locally ringed space isomorphic to an algebraic subset of ir for some n equipped with the sheaf of n valued regular functions cf 2 sect 3 2 denote by halgi m z 2 the subgroup of the homology group h i m z 2 which consists of all homology classes represented by zariski closed real algebraic hypersurfaces of m cf 2 sect 11 3 let h2ig m z 2 be the image of h g l m z 2 under the poincar6 duality isomorphism hm i m 7z 2 hi m z 2 the group h2ig m 7 2 is an important invariant with numerous interesting applications it has been recently extensively studied in 4 cf also the cited paper for a survey of earlier results and applications in this paper we investigate the group hlalg m z 2 where m is the underlying real algebraic variety of a complex projective nonsingular variety more precisely every complex projective variety v that is up to isomorphism an algebraic subset of cp n for some n endowed with the sheaf of c valued regular functions can be in the obvious way considered as a real algebraic variety and as such will be denoted by v of course dim v 2 dim v moreover v is an affine real algebraic variety cf 2 proposition 3 4 8 the very interesting and natural class of real varieties of type v has never been the subject of any serious investigation with a recent exception 3 7 7 if v is nonsingular then v is also nonsingular and we can consider the group hlalg v z 2 and in particular define d v dimz 2 h21g v e 2

stochastic solution of population balance equations for reactor networks this work presents a sequential modular approach to solve a generic network of reactors with a population balance model using a stochastic numerical method full coupling to the gas phase is achieved through operator splitting the convergence of the stochastic particle algorithm in test networks is evaluated as a function of network size recycle fraction and numerical parameters these test cases are used to identify methods through which systematic and statistical error may be reduced including by use of stochastic weighted algorithms the optimal algorithm was subsequently used to solve a one dimensional example of silicon nanoparticle synthesis using a multivariate particle model this example demonstrated the power of stochastic methods in resolving particle structure by investigating the transient and spatial evolution of primary polydispersity degree of sintering and tem style images an algorithm is presented to solve reactor networks with a population balance model a stochastic method is used to solve the population balance equations the convergence and efficiency of the reported algorithms are evaluated the algorithm is applied to simulate silicon nanoparticle synthesis in a 1d reactor particle structure is reported as a function of reactor length and time

real algebraic cycles on complex projective varieties if we identify cn with r2n then to any quasi affine complex algebraic variety v of dimension d corresponds in a natural way a real algebraic variety of dimension 2d in the sense of bcr which we will denote by vr since each complex morphism v v can be considered as a real morphism vr v r and all complex algebraic varieties are obtained by glueing affine pieces together we can extend this construction in a natural way to arbitrary complex algebraic varieties x in order to obtain xr the underlying real algebraic structure of x if f x x is a morphism of complex algebraic varieties then it is also in a natural way a regular mapping of real structures xr x r which by abuse of notation will be denoted by f as well if y is a real algebraic variety of dimension n and hk y z 2z is the singular homology group in dimension k with respect to the strong topology then for all k n there is a subgroup denoted by h alg k y z 2z consisting of the fundamental classes of k dimensional real algebraic subsets of y see bcr for a definition of the fundamental class of a real algebraic set and bk1 for more information on h alg k we will study the group h alg 2d 1 y z 2z for y xr where x is a nonsingular projective complex algebraic variety of dimension d in particular y is a nonsingular real algebraic variety of dimension n 2d that is compact for the strong topology it will be more convenient to consider h 1 alg y z 2z which is by definition the image of h alg n 1 y z 2z under the poincare duality isomorphism hn 1 y z 2z h 1 y z 2z then the homomorphism

vector bundles and regulous maps let x be a compact nonsingular affine real algebraic variety we prove that every pre algebraic vector bundle on x becomes algebraic after finitely many blowing ups using this theorem we then prove that the stiefel whitney classes of any pre algebraic mathbb r vector bundle on x are algebraic we also derive that the chern classes of any pre algebraic mathbb c vector bundles and the pontryagin classes of any pre algebraic mathbb r vector bundle are blow mathbb c algebraic we also provide several results on line bundles on x

the topology of real algebraic sets with isolated singularities in this paper we give a topological classification of real algebraic sets with isolated singularities showing that they are exactly smooth closed manifolds with smooth subpolyhedra crushed to points the question of which topological spaces are homeomorphic to real algebraic sets solutions of polynomial equations in euclidean space has been long studied in 1936 seifert showed that any smooth compact stably parallelizable manifold is diffeomorphic to a component of an algebraic set 121 and in 1952 nash extended this result to all smooth compact manifolds 11 in 1973 tognoli showed that any smooth compact manifold is diffeomorphic to a nonsingular algebraic set 13 so at least compact nonsingular algebraic sets are classified little has been done with singular algebraic sets however since the transversality arguments used by seifert nash and tognoli no longer apply except in some special cases one could use stability of singularities such as kuiper 71 and akbulut 1 used to show certain nonsmoothable pl manifolds are algebraic sets or one could use the projective version of seifert nash tognoli as king did 6 but one could still not hope these techniques would allow even a characterization of isolated singularities to get around this problem we take a cue from hironaka s resolution of singularities 4 the idea is to take a topological resolution of a space if it exists we can apply transversality techniques seifertnash tognoli to the resolved space and then blow down algebraically and end up with the original space as an algebraic set it seems likely that this technique allows one to classify all algebraic sets but in any case we show that it classifies all algebraic sets with isolated singularities in future papers we will use this technique to show for instance that all compact pl manifolds are homeomorphic to real algebraic sets 17 and that 2 dimensional real algebraic sets are topologically characterized as polyhedra satisfying sullivan s even local euler characteristic condition 16

a differentially weighted monte carlo method for two component coagulation the direct simulation monte carlo dsmc method for population balance modeling is capable of retaining the history of each simulation particle and is thus able to deal with multivariate properties in a simple and straightforward manner as opposed to conventional dsmc approaches that track equally weighted simulation particles a differentially weighted monte carlo method is extended to simulate two component coagulation processes and is thereby able to simulate the micromixing of the components a new feature of the method for this bivariate population balance modeling is that it is possible to specify how the simulation particles are distributed over the compositional axis this allows us to obtain information about particles in those regions of the size and composition distribution functions where the non weighted mc methods place insufficient simulation particles to obtain an inaccurate solution the new feature results in lower statistical noise for simulating two component coagulation which is validated by using two component coagulation cases for which analytical solutions exist a discrete process with sum kernel for initial monodisperse populations and a process with constant kernel for initial polydisperse populations

real algebraic morphisms represent few homotopy classes we study the problem of representing homotopy classes of maps between real algebraic varieties of regular maps

on real algebraic morphisms into even dimensional spheres given affine real algebraic varieties x and y let us denote by r x y the set of regular mappings real algebraic morphisms from x into y for definitions and notions of real algebraic geometry see 2 where the theory of real algebraic varieties is treated systematically our aim in this paper is to study the set m x s of regular mappings from affine nonsingular real algebraic varieties x into s the unit sphere ezn 1x2 1 earlier we obtained several results in this direction 3 4 cf also 2 chap 13 in particular it was shown in 4 that given a compact connected nonsingular orientable real algebraic subset x of rp of odd dimension k either each smooth mapping from x to sk is homotopic to a regular mapping or precisely those mappings of even topological degree have this property now we shall study the case of regular mappings into even dimensional spheres strangely enough the situation then is radically different from that mentioned above for odd dimensional spheres for a very large class of compact smooth manifolds of even dimension 2k most algebraic models x of these manifolds have the property that every regular mapping from x into s2k is null homotopic the meaning of most will be made precise later cf remark 1 6 theorem 2 1 example 2 3 this class of manifolds contains all compact co hypersurfaces of r2k 1

the linear process deferment algorithm a new technique for solving population balance equations in this paper a new stochastic algorithm for the solution of population balance equations is presented the population balance equations have the form of extended smoluchowski equations which include linear and source terms the new algorithm called the linear process deferment algorithm lpda is used for solving a detailed model describing the formation of soot in premixed laminar flames a measure theoretic formulation of a stochastic jump process is developed and the corresponding generator presented the numerical properties of the algorithm are studied in detail and compared to the direct simulation algorithm and various splitting techniques lpda is designed for all kinds of population balance problems where nonlinear processes cannot be neglected but are dominated in rate by linear ones in those cases the lpda is seen to reduce run times for a population balance simulation by a factor of up to 1000 with a negligible loss of accuracy

application of stochastic weighted algorithms to a multidimensional silica particle model highlights stochastic weighted algorithms swas are developed for a detailed silica model an implementation of swas with the transition kernel is presented the swas solutions converge to the direct simulation algorithm s dsa solution the efficiency of swas is evaluated for this multidimensional particle model it is shown that swas can be used for coagulation problems in industrial systems abstract this paper presents a detailed study of the numerical behaviour of stochastic weighted algorithms swas using the transition regime coagulation kernel and a multidimensional silica particle model the implementation in the swas of the transition regime coagulation kernel and associated majorant rates is described the silica particle model of shekar et al s shekar a j smith w j menz m sander m kraft a multidimensional population balance model to describe the aerosol synthesis of silica nanoparticles journal of aerosol science 44 2012 83 98 was used in conjunction with this coagulation kernel to study the convergence properties of swas with a multidimensional particle model high precision solutions were calculated with two swas and also with the established direct simulation algorithm these solutions which were generated using large number of computational particles showed close agreement it was thus demonstrated that swas can bemore successfully used with complex coagulation kernels and high dimensional particle models to simulate real world systems less

on approximating submanifolds by algebraic sets and a solution to the nash conjecture in n nash showed that any compact smooth submanifold m of ir is e isotopic to a union of connected components of a real algebraic variety if n 2 dim m e isotopic just means that there are arbitrarily small smooth isotopies to components of real algebraic varieties for smaller n however he was not able to get such a strong result but conjectured it was still true in w a proof of this conjecture was given which unfortunately had a gap later i and t2 gave correct proofs in the case where n is bigger than roughly 3 dim m 2 among other results in this paper we give a proof of this conjecture of nash in particular we will show

a hybrid particle number and particle model for efficient solution of population balance equations abstract this work presents a hybrid particle number and particle model to improve efficiency in solving population balance equations for type spaces spanning spherical and aggregate particles the particle number model tracks simpler spherical particles cheaply by storing only the number of particles with a given one dimensional internal coordinate while the particle model allows resolution of the detailed aggregate structure that occurs due to collision and coagulation between particles by storing distinct computational entries for each particle this approach is exact if primary particles are defined by their monomer count and the particle number model increments in single monomers a stochastic method is used to solve the population balance equations for the combined type space the hybrid method works well for large ensembles 2 12 particles with a detailed particle model where performing a finite number of particle number updates is demonstrated to be 40 50 cheaper than updating an equivalent ensemble of discrete particles these savings can be traded for a larger sample volume to increase the resolution in the particle size distribution or more repeat runs to reduce the total error run time improvements are curtailed at very high surface growth and coagulation rates due to the fixed cost of growth updates on the large aggregates formed however the hybrid method is still attractive in this case as its primary purpose is to reduce error by preventing saturation of the ensemble with simple particles at high inception rates

rival approaches to mathematical modelling in immunology in order to formulate quantitatively correct mathematical models of the immune system one requires an understanding of immune processes and familiarity with a range of mathematical techniques selection of an appropriate model requires a number of decisions to be made including a choice of the modelling objectives strategies and techniques and the types of model considered as candidate models the authors adopt a multidisciplinary perspective

rigidity and moduli space in real algebraic geometry let r be a real closed field and let x be an affine algebraic variety over r we say that x is universally map rigid umr for short if for each irreducible affine algebraic variety z over r the set of nonconstant rational maps from z to x is finite a bijective map open image in new window from an affine algebraic variety over r to x is called weak change of the algebraic structure of x if it is regular and 1 is a nash map which preserves nonsingular points we prove the following rigidity theorem every affine algebraic variety over r is umr up to a weak change of its algebraic structure let us introduce another notion let y be an affine algebraic variety over r we say that x and y are algebraically unfriendly if all the rational maps from x to y and from y to x are trivial i e zariski locally constant from the preceding theorem we infer that if dim x 1 then there exists a set open image in new window of weak changes of the algebraic structure of x such that for each t s r with t s open image in new window and open image in new window are algebraically unfriendly this result implies the following expected fact for each nonsingular affine algebraic variety x over r of positive dimension the natural nash structure of x does not determine the algebraic structure of x in fact the moduli space of birationally nonisomorphic nonsingular affine algebraic varieties over r which are nash isomorphic to x has the same cardinality of r this result was already known under the special assumption that r is the field of real numbers and x is compact and nonsingular

a stochastic weighted particle method for coagulation advection problems a spatially resolved stochastic weighted particle method for inception coagulation advection problems is presented convergence to a deterministic limit is briefly studied numerical experiments are carried out for two problems with very different coagulation kernels these tests show the method to be robust and confirm the convergence properties the robustness of the weighted particle method is shown to contrast with two direct simulation algorithms which develop instabilities

algebraic realization of manifolds with group actions abstract let g be a compact lie group and m a closed smooth g manifold with some restrictions on g or on the isotropy groups of the action we show that m is algebraically realized i e there exists a non singular real algebraic g variety which is equivariantly diffeomorphic to m

non recursive functions knots with thick ropes and self clenching thick hyperspheres we introduce an approach to certain geometric variational problems based on the use of the algorithmic unrecognizability of the n dimensional sphere for n 5 sometimes this approach allows one to prove the existence of infinitely many solutions of a considered variational problem this recursion theoretic approach is applied in this paper to a class of functionals on the space of c1 1 smooth hypersurfaces diffeomorphic to sn in rn 1 where n is any fixed number 5 the simplest of these functionals kv is defined by the formula kv n vol n 1 n r n where r n denotes the radius of injectivity of the normal exponential map for n rn l we prove the existence of an infinite set of distinct locally minimal values of kv on the space of c1 1 smooth topological hyperspheres in rn 1 for any n 5 the functional kv naturally arises when one attempts to generalize knot theory in order to deal with embeddings and isotopies of thick circles and more generally thick spheres into euclidean spaces we introduce the notion of knot with thick rope types the theory of knot with thick rope types turns out to be quite different from the classical knot theory because of the following result there exists an infinite set of non trivial knot with thick rope types in codimension one for every dimension greater than or equal to five

piecewise regular maps let v w be real algebraic varieties that is up to isomorphism real algebraic sets and x subseteq v some subset a map from x into w is said to be regular if it can be extended to a regular map defined on some zariski locally closed subvariety of v that contains x furthermore a continuous map f x rightarrow w is said to be piecewise regular if there exists a stratification mathscr s of v such that for every stratum s in mathscr s the restriction of f to each connected component of x cap s is a regular map by a stratification of v we mean a finite collection of pairwise disjoint zariski locally closed subvarieties whose union is equal to v assuming that the subset x of v is compact we prove that every continuous map from x into a grassmann variety or a unit sphere can be approximated by piecewise regular maps as an application we obtain a variant of the algebraization theorem for topological vector bundles if the variety v is compact and nonsingular we prove that each continuous map from v into a unit sphere is homotopic to a piecewise regular map of class mathscr c k where k is an arbitrary nonnegative integer

an efficient stochastic algorithm for studying coagulation dynamics and gelation phenomena a new class of efficient stochastic algorithms for the numerical treatment of coagulation processes is proposed the algorithms are based on the introduction of fictitious jumps combined with an acceptance rejection technique for distributions depending on particle size the increased efficiency is demonstrated by numerical experiments in particular gelation phenomena are studied

on real algebraic models of smooth manifolds an affine nonsingular real algebraic variety x diffeomorphic to a smooth manifold m is said to be an algebraic model of m the remarkable theorem of nash tognoli asserts that each compact smooth manifold m has an algebraic model 16 or 6 theorem 14 1 10 in fact there exists an infinite family x n of irreducible algebraic models of m such that xt and xj are birationally nonisomorphic for i j 10 cf also 7 for a proof in a special case in view of these results it seems natural and interesting to investigate algebro geometric properties of various algebraic models of a given smooth manifold this paper addresses a few questions of this type for notions and results of real algebraic geometry we refer the reader to the book 6 given a compact affine nonsingular real algebraic variety x denote by h x z 2 the subgroup of hk x z 2 of the homology classes represented by zariski closed fc dimensional algebraic subvarieties of x 6 chapter 11 or 11 let h x z 2 be the image of h k xfz 2 9 d dimx under the poincare duality isomorphism hd k x z 2 h x z 2 although the groups hlg x z 2 are one of the most important invariants of x a sample of applications can be found in 1 2 3 6 8 9 our knowledge of their behavior is still rather limited here we consider the following problem let m be a compact smooth manifold and let g be a subgroup of h m z 2 when are there an algebraic model x of m and a diffeomorphism c x m such that the induced isomorphism p h m z 2 h x z 2

a mixture model with dependent observations for the analysis of csfe labeling experiments recent advances in flow cytometry have resulted in the development of powerful bioassays to analyze the proliferation of cell populations the carboxy fluorescein diacetate succinimidyl ester cfse labeling experiment is one such assay that is widely used to study cell kinetics and as a standard tool for investigating lymphocyte proliferation several mathematical models have been proposed to describe cell proliferation during cfse experiments the statistical analysis of cfse labeling data has received little attention but it poses a number of methodological issues in this article we approach their analysis using a mixture model the mixing proportions are specified through an age dependent branching process that models the temporal organization of the cell population because the cfse molecules are partitioned between daughter cells when their mother divides the observations generated by this assay are dependent the data structure can be compared with that of a partially observed random effects mod

an efficient stochastic algorithm for simulating nano particle dynamics nano particles produced at high temperatures often undergo rapid coalescence with complex associated rate laws in this paper we develop and study the numerical properties of a stochastic algorithm for the modelling of nano particle dynamics in the free molecular regime following the work in a eibeck and w wagner 2000 siam j sci comput 22 3 802 821 we model the system as a markov process and introduce a new majorant kernel that enables us to extend the use of fictitious jumps to a wider class of problems we also include a source term we study the convergence properties of the algorithm the systematic error decreases as 1 n we then examine the efficiency of the algorithm by comparing it to a direct simulation monte carlo dsmc algorithm that described by d t gillespie 1975 j atmos sci 32 10 1977 1989 we also compare the efficiency of our new majorant with the linear majorant used in a eibeck and w wagner 2001 ann appl prob 11 4 1137 1165 the improved stochastic algorithm compares very favourably with the dsmc algorithm the cpu time required for simulation is orders of magnitude lower and for low particle numbers the cpu time increases linearly with particle number rather than as the square of the particle number as with the dsmc algorithm our majorant kernel enables us to simulate solutions to a wider variety of problems than the linear majorant and also gives a significant gain in efficiency the results of this report promise excellent efficiency of simulation of problems such as soot formation and synthesis of fumed silica and also for extension to a more general class of problems in which the population balance equation occurs

stochastic weighted particle methods for population balance equations with coagulation fragmentation and spatial inhomogeneity abstract this paper introduces stochastic weighted particle algorithms for the solution of multi compartment population balance equations in particular it presents a class of fragmentation weight transfer functions which are constructed such that the number of computational particles stays constant during fragmentation events the weight transfer functions are constructed based on systems of weighted computational particles and each of it leads to a stochastic particle algorithm for the numerical treatment of population balance equations besides fragmentation the algorithms also consider physical processes such as coagulation and the exchange of mass with the surroundings the numerical properties of the algorithms are compared to the direct simulation algorithm and an existing method for the fragmentation of weighted particles it is found that the new algorithms show better numerical performance over the two existing methods especially for systems with significant amount of large particles and high fragmentation rates

estimating division and death rates from cfse data the division tracking dye carboxyfluorescin diacetate succinimidyl ester cfse is currently the most informative labeling technique for characterizing the division history of cells in the immune system gett and hodgkin nat immunol 1 2000 239 244 have proposed to normalize cfse data by the 2 fold expansion that is associated with each division and have argued that the mean of the normalized data increases linearly with time t with a slope reflecting the division rate p we develop a number of mathematical models for the clonal expansion of quiescent cells after stimulation and show within the context of these models under which conditions this approach is valid we compare three means of the distribution of cells over the cfse profile at time t the mean t the mean of the normalized distribution 2 t and the mean of the normalized distribution excluding nondivided cells 2 t in the simplest models which deal with homogeneous populations of cells with constant division and death rates the normalized frequency distribution of the cells over the respective division numbers is a poisson distribution with mean 2 t pt where p is the division rate the fact that in the data these distributions seem gaussian is therefore insufficient to establish that the times at which cells are recruited into the first division have a gaussian variation because the poisson distribution approaches the gaussian distribution for large pt excluding nondivided cells complicates the data analysis because 2 t pt and only approaches a slope p after an initial transient in models where the first division of the quiescent cells takes longer than later divisions all three means have an initial transient before they approach an asymptotic regime which is the expected t 2pt and 2 t 2 t pt such a transient markedly complicates the data analysis after the same initial transients the normalized cell numbers tend to decrease at a rate e dt where d is the death rate nonlinear parameter fitting of cfse data obtained from gett and hodgkin to ordinary differential equation ode models with first order terms for cell proliferation and death gave poor fits to the data the smith martin model with an explicit time delay for the deterministic phase of the cell cycle performed much better nevertheless the insights gained from analysis of the odes proved useful as we showed by generating virtual cfse data with a simulation model where cell cylce times were drawn from various distributions and then computing the various mean division numbers

on the integral hodge conjecture for real varieties i we formulate the real integral hodge conjecture a version of the integral hodge conjecture for real varieties and raise the question of its validity for cycles of dimension 1 on uniruled and calabi yau threefolds and on rationally connected varieties we relate it to the problem of determining the image of the borel haefliger cycle class map for 1 cycles with the problem of deciding whether a real variety with no real point contains a curve of even geometric genus and with the problem of computing the torsion of the chow group of 1 cycles of real threefolds new results about these problems are obtained along the way

algebraic cycles and approximation theorems in real algebraic geometry let af be a compact c manifold a theorem of nash tognoli asserts that m has an algebraic model that is m is diffeomorphic to a nonsingular real algebraic set x let fv afx z 2 denote the subgroup of hk x z 2 of the cohomology classes determined by algebraic cycles of codi mension k on x assuming that m is connected orientable and dim m 5 we prove in this paper that a subgroup g of h2 m z 2 is isomorphic to h x z 2 for some algebraic model x of m if and only if w2 tm is in g and each element of g is of the form w2k for some real vector bundle over m where w2 stands for the second stiefel whitney class a result of this type was previously known for subgroups g of hx m z 2

a resolution theorem for homology cycles of real algebraic varieties the theorem is the best possible in the sense that the components of nonsingular algebraic subsets in the conclusion cannot be replaced by nonsingular algebraic subsets see remark 14 it is well known that the topology of the singularities of 7 2 homology cycles is closely related to real algebraic structures on v ak 2 ak3 therefore it is natural to study to what extent general homology cycles differ from the cycles induced by submanifolds the main theorem of this paper implies that all the singularities of homology cycles can be smoothed by a blowing up process this result should be viewed as a homology version of the resolution theorem h the reason this result is not an easy consequence of h is that not all homology cycles of v may be represented by algebraic subsets there are real obstructions e g ak2 even if they were the map resolving them may introduce new bad homology cycles upstairs the result turns out to be a useful device in topologically classifying real algebraic sets ak6 it also gives interesting corollaries in algebraic topology ak2 another result of this paper theorem 4 says that any smooth map f w v between nonsingular algebraic sets can be approximated by a rational map after changing w by a rational diffeomorphism if the homotopy class of f contains a rational representative this strengthens proposition 2 3 of aks for given algebraic sets vcf and w e n m a function f v w is called an entire rational function if f x p x q x where p ir and q ir ir

algebraic models of smooth manifolds in fact we conjecture that such a family can be chosen uncountable in view of this theorem it seems natural and interesting to investigate algebrogeometric properties of various algebraic models of a given smooth manifold let us now describe one of these problems which will be the main object of study in this paper for notions definitions and results of real algebraic geometry we refer the reader to the book 9 given a compact nonsingular real affine algebraic variety x denote by h ig x 7z 2 the subgroup of nk x z 2 of homology classes represented by zariski closed k dimensional algebraic subvarieties of x cf 9 chap 1 1 let h lg x z 2 be the image of h a k x 71 2 d dimx under the poincar6 duality isomorphism hd k x 71 2 o hk x 71 2 these groups h ig x 71 2 are undoubtedly one of the most important invariants of an algebraic variety x still our knowledge of the behavior of h g x 71 2 is rather limited the main open problem is the following

the weierstrass approximation theorem for maps between real algebraic varieties p n r for some n endowed with the zariski topology and the sheaf of rvalued regular functions morphisms between real algebraic varieties will be called regular maps every real algebraic variety carries also the euclidean topology that is the topology induced by the usual metric topology on r unless explicitly stated otherwise all topological notions related to real algebraic varieties will refer to the euclidean topology given two real algebraic varieties x and y we regard the set r x y of all regular maps from x into y as a subset of the space c x y of all continuous maps fromx into y endowed with the compact open topology if x and y are nonsingular then we also regard r x y as a subset of the space c x y of all c maps from x into y endowed with the c compact open topology the weak c topology in the terminology used in 6