continuous functions on the plane regular after one blowing up we study rational functions admitting a continuous extension to the real affine space first of all we focus on the regularity of such functions exhibiting some nice properties of their partial derivatives afterwards since these functions correspond to rational functions which become regular after some blowings up we work on the plane where it suffices to blow up points and then we can count the number of stages of blowings up necessary in the latest parts of the paper we investigate the ring of rational continuous functions on the plane regular after one stage of blowings up in particular we prove a positivstellensatz without denominator in this ring

applications of soft sets in ideal theory of bck bci algebras the notion of soft ideals and idealistic soft bck bci algebras is introduced and several examples are given relations between soft bck bci algebras and idealistic soft bck bci algebras are provided the intersection union and operation and or operation of soft ideals and idealistic soft bck bci algebras are established

soft sets and soft groups molodtsov introduced the concept of soft set theory which can be used as a generic mathematical tool for dealing with uncertainty in this paper we introduce the basic properties of soft sets and compare soft sets to the related concepts of fuzzy sets and rough sets we then give a definition of soft groups and derive their basic properties using molodtsov s definition of the soft sets

sums of two squares in analytic rings we study analytic singularities for which every positive semidefinite analytic function is a sum of two squares of analytic functions this is a basic useful property of the plane but difficult to check in other cases in particular what about z 2 xy z 2 yx 2 y 3 z 2 x 3 y 4 or z 2 x 3 xy 3 in fact the unique positive examples we can find are the brieskorn singularity the union of two planes in 3 space and the whitney umbrella conversely we prove that a complete intersection with that property other than the seven embedded surfaces already mentioned must be a very simple deformation of the two latter namely in particular except for the stems z 2 x 2 and z 2 yx 2 all singularities are real rational double points

approximation by piecewise regular maps abstract a real algebraic variety w of dimension m is said to be uniformly rational if each of its points has a zariski open neighborhood which is biregularly isomorphic to a zariski open subset of r m let l be any nonnegative integer we prove that every map of class c l from a compact subset of a real algebraic variety into a uniformly rational real algebraic variety can be approximated in the c l topology by piecewise regular maps of class c k where k is an arbitrary integer satisfying k l next we derive consequences regarding algebraization of topological vector bundles

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

approximation of maps into spheres by piecewise regular maps of class c k the aim of this paper is to prove that every continuous map from a compact subset of a real algebraic variety into a sphere can be approximated by piecewise regular maps of class mathcal c k where k is an arbitrary nonnegative integer

complex cycles on algebraic models of smooth manifolds every compact smooth manifold m is diffeomorphic to the set x mathbb r of real points of a nonsingular projective real algebraic variety x which is called an algebraic model of m each algebraic cycle of codimension k on the complex variety x mathbb c x times mathbb r mathbb c determines a cohomology class in h 2k x mathbb r mathbb d where mathbb d denotes mathbb z or mathbb q we investigate the behavior of such cohomology classes as x runs through the class of algebraic models of m

generalised fuzzy soft sets in this paper we define generalised fuzzy soft sets and study some of their properties application of generalised fuzzy soft sets in decision making problem and medical diagnosis problem has been shown

characterizations of ordered semigroups by the properties of their fuzzy ideals in this paper we characterize different classes of ordered semigroups by the properties of their ideals quasi ideals and bi ideals we also characterize these classes by the properties of their fuzzy ideals fuzzy quasi ideals and fuzzy bi ideals

curve rational functions let w be a subset of the set of real points of a real algebraic variety x we investigate which functions f w rightarrow mathbb r are the restrictions of rational functions on x we introduce two new notions curve rational functions i e continuous rational on algebraic curves and arc rational functions i e continuous rational on arcs of algebraic curves we prove that under mild assumptions the following classes of functions coincide continuous hereditarily rational introduced recently by the first named author curve rational and arc rational in particular if w is semialgebraic and f is arc rational then f is continuous and semialgebraic we also show that an arc rational function defined on an open set is arc analytic i e analytic on analytic arcs furthermore we study rational functions on products of varieties as an application we obtain a characterization of regular functions finally we get analogous results in the framework of complex algebraic varieties

continuous rational maps into spheres let x be a compact nonsingular real algebraic variety we prove that if a continuous map from x into the unit p sphere is homotopic to a continuous rational map then under certain assumptions it can be approximated in the compact open topology by continuous rational maps as a byproduct we also obtain some results on approximation of smooth submanifolds by nonsingular subvarieties

algebraic structures of soft sets associated with new operations recently new operations have been defined for soft sets in this paper we study some important properties associated with these new operations a collection of all soft sets with respect to new operations give rise to four idempotent monoids then with the help of these monoids we can study semiring hemiring structures of soft sets some of these semirings hemirings are actually lattices finally we show that soft sets with a fixed set of parameters are mv algebras and bck algebras

semi algebraic geometry with rational continuous functions let x be a real algebraic subset of mathrm mathbb r n we investigate the theory of algebraically constructible functions on x and the description of the semi algebraic subsets of x when we replace the polynomial functions on x by some rational continuous functions on x

soft semirings molodtsov introduced the concept of soft sets which can be seen as a new mathematical tool for dealing with uncertainty in this paper we initiate the study of soft semirings by using the soft set theory the notions of soft semirings soft subsemirings soft ideals idealistic soft semirings and soft semiring homomorphisms are introduced and several related properties are investigated

on some new operations in soft set theory molodtsov introduced the theory of soft sets which can be seen as a new mathematical approach to vagueness in this paper we first point out that several assertions proposition 2 3 iv vi proposition 2 4 and proposition 2 6 iii iv in a previous paper by maji et al p k maji r biswas a r roy soft set theory comput math appl 45 2003 555 562 are not true in general by counterexamples furthermore based on the analysis of several operations on soft sets introduced in the same paper we give some new notions such as the restricted intersection the restricted union the restricted difference and the extended intersection of two soft sets moreover we improve the notion of complement of a soft set and prove that certain de morgan s laws hold in soft set theory with respect to these new definitions

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

dynamics as a mechanism preventing the formation of finer and finer microstructure we study the dynamics of pattern formation in the one dimensional partial differential equation u u w u x x u xxt u 0 text u u x t text x in 0 1 text t 0 proposed recently by ball holmes james pego swart bhjps as a mathematical cartoon for the dynamic formation of microstructures observed in various crystalline solids here w is a double well potential like 1 4 ux 2 1 2 what makes this equation interesting and unusual is that it possesses as a lyapunov function a free energy consisting of kinetic energy plus a nonconvex elastic energy but no interfacial energy contribution which does not attain a minimum but favours the formation of finer and finer phase mixtures e u u t int limits 0 1 frac u t 2 2 w u x frac u 2 2 dx our analysis of the dynamics confirms the following surprising and striking difference between statics and dynamics conjectured in bhjps on the basis of numerical simulations of swart holmes sh while minimizing the above energy predicts infinitely fine patterns mathematically weak but not strong convergence of all minimizing sequences unvn of e u v in the sobolev space w1p 0 1 l2 0 1 solutions to the evolution equation of ball et al typically develop patterns of small but finite length scale mathematically strong convergence in w1p 0 1 l2 0 1 of all solutions u t ut t with low initial energy as time t

soft ideals and soft filters of soft ordered semigroups it is shown that soft ordered semigroups over an ordered semigroup are actually soft ordered subsemigroups of s a soft idealistic semigroups over an ordered semigroup are soft ideals of soft ordered semigroup s a the concept of soft filters of a soft ordered semigroup is introduced it is shown that restricted complement of a soft filter is a soft prime ideal of s a

attainment results for nematic elastomers we consider a class of non quasiconvex frame indifferent energy densities which includes ogden type energy densities for nematic elastomers for the corresponding geomet rically linear problem we provide an explicit minimizer of the energy functional satisfying a nontrivial boundary condition other attainment results both for the nonlinear and the lin earized model are obtained by using the theory of convex integration introduced by muller and u sverak in the context of crystalline solids

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 euler equations as a differential inclusion we propose a new point of view on weak solutions of the euler equations describing the motion of an ideal incompressible fluid in r n with n 2 we give a reformulation of the euler equations as a differential inclusion and in this way we obtain transparent proofs of several celebrated results of v scheffer and a shnirelman concerning the non uniqueness of weak solutions and the existence of energy decreasing solutions our results are stronger because they work in any dimension and yield bounded velocity and pressure