sur une conjecture de kato et kuzumaki concernant les hypersurfaces de fano resume nous montrons que le corps qp et les corps de nombres totalement imaginaires verifient la propriete c 1 conjecturee par kato et kuzumaki en 1986 autrement dit si k est l un de ces corps et f k x0 xn est un polynome homogene de degre d 6 n tout element de k s ecrit comme produit de normes depuis des extensions finies de k dans lesquelles f admet un zero non trivial nous etablissons aussi la conjecture d ax sur les corps pseudo algebriquement clos parfaits pour les corps dont le groupe de galois absolu est un pro p groupe

singularities in vibro impact dynamics abstract the non differentiable nature of vibro impact dynamics can lead to breakdown of the global stable manifold theorem applicable to smooth dynamical systems in one dimensional periodically excited system the breakdown leads to the shredding of the stable manifolds and this process is analyzed in detail bodily translation of filaments of stable manifolds can lead to homoclinic intersections and this topic is discussed in the context of a new bifurcation due to re entrance shredding leads to a strong similarity between the geometries of the singular subspaces and the geometries of homoclinic tangles and strange attracting sets

limit cycles bifurcate from centers of discontinuous quadratic systems like for smooth quadratic systems it is important to determine the maximum order of a fine focus and the cyclicity of discontinuous quadratic systems previously examples of discontinuous quadratic systems with five limit cycles bifurcated from a fine focus of order 5 have been constructed in this paper we construct a class of discontinuous quadratic systems with a fine focus of order 9 in addition by using a method similar to that developed by c christopher for smooth systems which allows one to estimate the cyclicity just from the lower order terms of lyapunov constants we show that the cyclicity of discontinuous quadratic systems is at least 9 thus improving on previous results

on the kernel of the reciprocity map of varieties over local fields this thesis is about higher dimensional class field theory of varieties over local and finite fields their abelianized etale fundamental group is compared with an attached class group via the reciprocity map like in the classical case of a local or global field the main results in this thesis are about the structure of the kernel of their reciprocity maps and criteria for the vanishing of it resp

groupe de chow de codimension deux des varietes definies sur un corps de nombres un theoreme de finitude pour la torsion letx be a smooth projective variety defined over a number field and let ch2 x denote the chow group of codimension two cycles modulo rational equivalence we show that if the cohomology grouph2 x ox vanishes then the torsion subgroup of ch2 x is a finite group this result covers all previous results in this direction the hypothesish2 x ox 0 is used to lift line bundles

bifurcations of dynamical systems with sliding derivation of normal form mappings this paper is concerned with the analysis of so called sliding bifurcations in n dimensional piecewise smooth dynamical systems with discontinuous vector field these novel bifurcations occur when the system trajectory interacts with regions on the discontinuity set where sliding is possible the derivation of appropriate normal form maps is detailed it is shown that the leading order term in the map depends on the particular bifurcation scenario considered this is in turn related to the possible bifurcation scenarios exhibited by a periodic orbit undergoing one of the sliding bifurcations discussed in the paper a third order relay system serves as a numerical example

class field theory for arithmetic schemes let x be a regular arithmetic scheme i e a regular integral separated scheme flat and of finite type over spec mathbb z generalising classical class field theory for number fields we define a class group cx and show there is a natural surjective map c x rightarrow pi 1 ab x whose kernel is the connected component of 0

r equivalence in adjoint classical groups over fields of virtual cohomological dimension 2 let f be a field of characteristic not 2 whose virtual cohomological dimension is at most 2 let g be a semisimple group of adjoint type defined over f let rg f denote the normal subgroup of g f consisting of elements r equivalent to identity we show that if g is of classical type not containing a factor of type d n g f rg f 0 if g is a simple classical adjoint group of type d n we show that if f and its multi quadratic extensions satisfy strong approximation property then g f rg f 0 this leads to a new proof of the r triviality of f rational points of adjoint classical groups defined over number fields

fuzzy topology on fuzzy sets product fuzzy topology and fuzzy topological groups the notion of product fuzzy topology in the case of fuzzy topology on fuzzy sets is introduced and the product invariance of fuzzy hausdorffness compactness connectedness are examined the product fuzzy topology is used to define fuzzy group topology on a fuzzy subgroup of a group g and some properties of fuzzy topological groups are obtained

the arithmetic of zero cycles on surfaces with geometric genus and irregularity zero let x be a smooth projective geometrically irreducible surface over a perfect field f throughout this paper it will be assumed that the geometric genus pg and the irregularity q of x both vanish denote the separable closure of f by f let x x be the surface obtained from x by base extension it will also be assumed that the group ao x of rational equivalence classes of zero cycles of degree zero on x vanishes this is a technical hypothesis which could presumably be eliminated at the expense of working with ker ao x ao x for want of a better name and for ease in stating various results any surface which satisfies these three hypotheses will be called a pseudo rational surface bloch 2 has conjectured that the vanishing of ao x should follow from the assumption that po q 0 this was proven by bloch kas and lieberman 5 for all such surfaces which are not of general type i e which have kodaira dimension less than 2 it has also been proven for particular surfaces of general type by inose and mizukami 23 barlow 1 and keum 25 consequently the class of pseudo rational surfaces includes rational surfaces enriques surfaces elliptic surfaces with q 0 the classical godeaux surface burniat inoue surfaces campedelti surfaces and the surfaces of barlow and of keum this paper will study ao x for pseudo rational surfaces defined over fields of number theoretic interest bloch 4 introduced k theoretic techniques into the study of zero cycles on rational surfaces his work was extended 7 12 15 27 28 32 36 to achieve a thorough understanding of such cycles colliot th616ne and raskind 10 developed this machinery further to study codimension two cycles on any variety the author 14 applied these techniques to enriques surfaces raskind 30 used them to study zero cycles on pseudo rational surfaces one of the main results of this paper is a new proof of the following theorem of raskind 31

non abelian class field theory for arithmetic surfaces let open image in new window be a regular arithmetic surface assume that for all irreducible curves open image in new window there are given open normal subgroups of 1 c which fulfill a compatibility condition at all closed points x open image in new window we then show that these data uniquely determine a normal subgroup of 1 open image in new window this is used to construct abelian class field theory for arithmetic surfaces using only k0 and k1 groups of local and global fields

suppression of period doubling in the dynamics of a bouncing ball abstract we present a theoretical and experimental study of a bouncing ball system where the effect of a near resonant perturbation is to suppress the onset of the first period doubling bifurcation near the bifurcation point the full dynamical equations are reduced to a discrete time map governing the dynamics on a slowly oscillating center manifold the derivation emphasizes the geometry of the phase space dynamics and serves to clarify several points of a recently proposed theory regarding the effects of strong near resonant perturbations our experimental results agree with the simple theory for moderate perturbations for larger perturbations higher order effects must be included

a periodically forced piecewise linear oscillator a single degree of freedom non linear oscillator is considered the non linearity is in the restoring force and is piecewise linear with a single change in slope such oscillators provide models for mechanical systems in which components make intermittent contact a limiting case in which one slope approaches infinity an impact oscillator is also considered harmonic subharmonic and chaotic motions are found to exist and the bifurcations leading to them are analyzed

unramified class field theory of arithmetical surfaces let k be an algebraic number field 9k its ring of integers and v a non empty open subscheme of spec ck let x be a projective smooth geometrically irreducible scheme over k and x a regular proper flat scheme over v such that x x vk x the purpose of this paper is to give an explicit description of the abelian fundamental groups ab x and 7t b x using some idele class groups attached to x and x by a k theoretical method its outline is as follows in general for a noetherian scheme z and i 0 or 1 we define the group ski z to be the cokernel of

dynamics and bifurcations of nonsmooth systems a survey abstract in this survey we discuss current directions of research in the dynamics of nonsmooth systems with emphasis on bifurcation theory an introduction to the state of the art also for non specialists is complemented by a presentation of the main open problems we illustrate the theory by means of elementary examples the main focus is on piecewise smooth systems which have recently attracted a lot of attention but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities this extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context a dedicated section is devoted to concrete applications that stimulate the development of the field this survey is concluded by an extensive bibliography

riddling and chaotic synchronization of coupled piecewise linear lorenz maps abstract we investigate the parametric evolution of riddled basins related to synchronization of chaos in two coupled piecewise linear lorenz maps riddling means that the basin of the synchronized attractor is shown to be riddled with holes belonging to another basin in an arbitrarily fine scale which has serious consequences on the predictability of the final state for such a coupled system we found that there are wide parameter intervals for which two piecewise linear lorenz maps exhibit riddled basins globally or locally which indicates that there are riddled basins in coupled lorenz equations as previously suggested by numerical experiments the use of piecewise linear maps makes it possible to prove rigorously the mathematical requirements for the existence of riddled basins

fuzzy topology on fuzzy sets and tolerance topology in this paper fuzzy topology has been defined on a fuzzy set the notion of quasi coincidence is extended in order to fit this new situation and to generalize the concept of q neighbourhood introduced originally by ming and ming 1980 the generalized q neighbourhood system has been utilized to introduce tolerance topology on a fuzzy set with a fuzzy tolerance relation

low dimensional maps for piecewise smooth oscillators abstract dynamics of the piecewise smooth nonlinear oscillators is considered for which general methodology of reducing multidimensional flows to low dimensional maps is proposed this includes a definition of piecewise smooth oscillator and creation of a global iterative map providing an exact solution the global map is comprised of local maps which are constructed in the smooth sub regions of phase space to construct this low dimensional map it is proposed to monitor the points of intersections of a chosen boundary between smooth subspaces by a trajectory the dimension reduction is directly related to the dimension of the chosen boundary and the lower its dimension is the larger dimension reduction can be achieved full details are given for a drifting impact oscillator where the five dimensional flow is reduced to one dimensional 1d approximate analytical map first an exact two dimensional map has been formulated and analysed a further reduction to 1d approximate map is introduced and discussed standard nonlinear dynamic analysis reveals a complex behaviour ranging from periodic oscillations to chaos and co existence of multiple attractors accuracy of the constructed maps is examined by comparing with the exact solutions for a wide range of the system parameters

a generalization of serre s condition mathrm f f with applications to the finiteness of unramified cohomology in this paper we introduce a condition mathrm f m on a field k for a positive integer m that generalizes serre s condition f and which still implies the finiteness of the galois cohomology of finite galois modules annihilated by m and algebraic k tori that split over an extension of degree dividing m as well as certain groups of etale and unramified cohomology various examples of fields satisfying mathrm f m including those that do not satisfy f are given

periodic solution finder for an impact oscillator with a drift in this paper an efficient semi analytical method is developed to compute periodic solutions for a new model of an impact oscillator with a drift which explains the progression mechanism in vibro impact systems and can be used to optimize their performance the method constructs a periodic response assuming that each period is comprised of a sequence of distinct phases for which analytical solutions are known for example a period may consist of the following sequential phases i contact with progression ii contact without progression iii no contact and iv contact without progression using this information a system of four piecewise linear first order differential equations is transformed to a system of non linear algebraic equations the method allows one to accurately predict a range of control parameters for which the best progression rates are obtained

heteroclinic bifurcations in damped rigid block motion it is shown that heteroclinic bifurcations are present in a piecewise linear system of ordinary differential equations that describe the rocking motion of a slender rigid block with damping an exact expression is given for the bifurcation amplitude stable and unstable manifolds are analytically extended to explicitly reveal the intersections as the damping increases these bifurcations occur only at increasingly large forcing amplitudes as manifolds move further apart no perturbation methods are used in this analysis

a finiteness theorem for zero cycles over p adic fields let r be a henselian discrete valuation ring let x be a regular projective flat scheme over spec r with generalized semistable reduction we prove a bijectivity theorem for etale cycle class maps of the chow group of 1 cycles on x as an application we prove a finiteness theorem for the chow group of 0 cycles on a projective smooth variety over a p adic field

normal form maps for grazing bifurcations in n dimensional piecewise smooth dynamical systems abstract this paper presents a unified framework for performing local analysis of grazing bifurcations in n dimensional piecewise smooth systems of odes these occur when a periodic orbit has a point of tangency with a smooth n 1 dimensional boundary dividing distinct regions in phase space where the vector field is smooth it is shown under quite general circumstances that this leads to a normal form map that contains to lowest order either a square root or a 3 2 type singularity according to whether the vector field is discontinuous or not at the grazing point in particular contrary to what has been reported in the literature piecewise linear local maps do not occur generically first the concept of a grazing bifurcation is carefully defined using appropriate non degeneracy conditions next complete expressions are derived for calculating the leading order term in the normal form poincare map at a grazing bifurcation point in arbitrary systems using the concept of a discontinuity mapping finally the theory is compared with numerical examples including bilinear oscillators a relay feedback controller and general third order systems

class field theory for open curves over p adic fields we introduce the idele class group for quasi projective curves over p adic fields and show that the kernel of the reciprocity map is divisible this extends saito s class field theory for projective curves saito in j number theory 21 44 80 1985

border collision bifurcations in two dimensional piecewise smooth maps recent investigations on the bifurcations in switching circuits have shown that many atypical bifurcations can occur in piecewise smooth maps that cannot be classified among the generic cases like saddle node pitchfork or hopf bifurcations occurring in smooth maps in this paper we first present experimental results to establish the need for the development of a theoretical framework and classification of the bifurcations resulting from border collision we then present a systematic analysis of such bifurcations by deriving a normal form the piecewise linear approximation in the neighborhood of the border we show that there can be eleven qualitatively different types of border collision bifurcations depending on the parameters of the normal form and these are classified under six cases we present a partitioning of the parameter space of the normal form showing the regions where different types of bifurcations occur this theoretical framework will help in explaining bifurcations in all systems which can be represented by two dimensional piecewise smooth maps

class field theory for a product of curves over a local field we prove that the kernel of the reciprocity map for a product of curves over a p adic field with split semi stable reduction is divisible we also consider the k1 of a product of curves over a number field

fuzzy topology on fuzzy sets fuzzy semicontinuity and fuzzy semiseparation axioms the concept of a fuzzy topology on a fuzzy set was introduced in fuzzy set syst 45 1992 103 the aim of this paper is to introduce fuzzy semicontinuity and fuzzy semiseparation axioms in this new situation and validity of some characterization of these concepts have been examined also we define a fuzzy generalized semiopen set and introduce fuzzy separation axioms by using this concept fuzzy semiconnected and fuzzy semicompact spaces are introduced and some of their properties are discussed

tame class field theory for arithmetic schemes takagi s class field theory gave a decription of the abelian extensions of a number field k in terms of ideal groups in k in the 1980s it k kato and it s saito global class field theory of arithmetic schemes applications of algebraic k theory to algebraic geometry and number theory proc ams ims siam joint summer res conf boulder colo 1983 part i contemp math 55 255 331 1986 zbl 0614 14001 were able to generalize class field theory to higher dimensional fields and to describe their abelian extensions using a generalized idele class group whose definition is quite involved in the case of unramified extensions however the class fields can be described geometrically using chow groups for fields of positive characteristic a similarly geometric description for tamely ramified extensions was obtained by the author and it m spiess j reine angew math 527 13 36 2000 zbl 0961 14013 in this article an analogous result is proved for the case of mixed characteristic

border collision bifurcations including period two to period three for piecewise smooth systems abstract we examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter in the simplest case there is a surface in phase space along which the map has no derivative or has two one sided derivatives is the border of two regions in which the map is smooth as the parameter is varied a fixed point e may collide with the border and we may assume that this collision occurs at 0 a variety of bifurcations occur frequently in such situations but never or almost never occur in smooth systems in particular e may cross the border and so will exist for 0 but it may be a saddle in one case say 0 for 0 there may be a stable period 3 orbit which similarly shrinks to e0 as 0 hence one observes the following stable periodic orbits a stable period 2 orbit collapses to a point and is reborn as a stable period 3 orbit we also see analogously stable period 2 to stable period p orbit bifurcations with p 5 11 52 or period 2 to quasi periodic or even to a chaotic attractor we believe this phenomenon will be seen in many applications

cycles on curves over global fields of positive characteristic let k be a global field of positive characteristic and let x spec k be a smooth projective curve we study the zero dimensional cycle group v x ker sik 1 x k 1 k and the one dimensional cycle group w x coker k 2 k h 0 zar x k 2 addressing the conjecture that v x is torsion and w x is finitely generated the main idea is to use abhyankar s theorem on resolution of singularities to relate the study of these cycle groups to that of the k groups of a certain smooth projective surface over a finite field

some consequences of wiesend s higher dimensional class field theory we use a result of g wiesend to establish the relation between the integral singular homology in degree zero and the abelianized tame fundamental group of a regular connected scheme of finite type over spec mathbb z

a nonsmooth coulomb friction oscillator abstract a forced coulomb friction oscillator whose frictional force is allowed to vary with displacement is analyzed geometrically the equation of motion for the oscillator is piecewise linear we geometrically observe the nature of the flow in each region of solvability and then see how these solutions interact at the boundary of the regions the dynamics of the flow is viewed in terms of a map on the boundary between the regions for chaotic motion we geometrically construct the strange attractor and show that its exact behavior is that of a one dimensional map the following dynamical properties arise from the nonsmooth nature of the coulomb friction law the flow may not be invertible the flow may reach its attractor in finite time the dimension of the attractor may be less than or equal to two embeddings of an observable may not be diffeomorphic to the full phase flow

on the origin and bifurcations of stick slip oscillations a recently proposed model of macroscopic friction is investigated using methods of dynamical systems analysis particular emphasis is put on the bifurcations associated with the appearance of stick

global dynamics of a vibro impacting linear oscillator abstract the steady state vibro impacting responses of one dimensional harmonically excited linear oscillators are studied by using a modern dynamical systems approach allied with numerical simulation the steady state motions are attracting sets in the system phase space and capture initial conditions in their domains of attraction unlike the free harmonically excited oscillator the phase space of a vibro impacting system may be inhabited by many attracting sets for example there are sub harmonic multi impact periodic orbits and chaotic steady state responses in order to build a qualitative understanding of vibro impact response an attempt is made to build generic topological models of their phase spaces for physically significant parameter ranges use is made of the poincare section or stroboscopic mapping technique essentially following an initial impact forwards or backwards in time to subsequent or previous impacts using a computer the qualitative understanding gained from the analysis and simulations is discussed in an engineering context

zero cycles sur les fibrations au dessus d une courbe de genre quelconque resume soit x une variete propre et lisse sur un corps de nombres k des conjectures sur l image du groupe de chow des zero cycles de x dans le produit des memes groupes sur tous les completes de k ont ete proposees par colliot thelene kato et saito nous demontrons ces conjectures pour l espace total de fibrations en varietes rationnellement connexes verifiant l approximation faible au dessus de courbes dont le groupe de tate shafarevich est fini sous une hypothese d abelianite sur les fibres singulieres

two dimensional map for impact oscillator with drift an impact oscillator with a drift is considered using a simple co ordinates transformation the bounded oscillations are separated from the drift in general the dynamic state of the system is fully described by four variables time relative displacement p and velocity y of the mass and relative displacement of the slider top q however this number can be reduced by two if the beginning of the progression phase is monitored in this paper a new two dimensional numerical map is developed and its dynamics is discussed

some results on fuzzy topology on fuzzy sets abstract the object of the present paper is to introduce several fundamental concepts in the context of fuzzy topology on fuzzy sets introduced by chakrabarty and ahsanullah 2 the validity of some standard characterizations of fuzzy continuous proper functions have been examined the concept of hausdorffness regularity normality complete normality compactness connectedness are introduced and some interesting results are obtained

voevodsky s motives and weil reciprocity we describe somekawa s k group associated to a finite collection of semiabelian varieties or more general sheaves in terms of the tensor product in voevodsky s category of motives while somekawa s definition is based on weil reciprocity voevodsky s category is based on homotopy invariance we apply this to explicit descriptions of certain algebraic cycles

galois equivariance and stable motivic homotopy theory for a finite galois extension of fields l k with galois group g we study a functor from the g equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical galois correspondence we show that after completing at a prime and the motivic hopf map this results in a full and faithful embedding whenever k is real closed and l k i it is a full and faithful embedding after completion if a motivic version of serre s finiteness theorem is valid we produce strong necessary conditions on the field extension l k for this functor to be full and faithful along the way we produce several results on the stable c2 equivariant betti realization functor and prove convergence theorems for the p primary c2 equivariant adams spectral sequence

non linear dynamic interactions of a jeffcott rotor with preloaded snubber ring a two degrees of freedom model of a jeffcott rotor with a preloaded snubber ring subjected to out of balance excitation has been developed the purely impact interactions have been investigated the rotor makes intermittent contacts with the preloaded snubber ring and as a consequence it can be in one of five different contact regimes which boundaries have been found analytically the current location of the snubber ring has been determined using the principle of the minimum elastic energy in the snubber ring consequently a non linear piecewise smooth dynamical system has been obtained and studied numerically the results in form of bifurcation diagrams phase portraits and poincare maps show significant differences for the cases with and without preloading

non periodic motion caused by grazing incidence in an impact oscillator the motion of a single degree of freedom periodically forced oscillator subjected to a rigid amplitude constraint is considered using analytical methods the singularities caused by grazing impact are studied it is shown that as a stable periodic orbit comes to grazing impact under the control of a single parameter a special type of bifurcation occurs the motion after the bifurcation may be non periodic and a criterion for this based on orientation and eigenvalues is given

topology of vibro impact systems in the neighborhood of grazing abstract the grazing bifurcation is considered for the newtonian model of vibro impact systems a brief review on the conditions sufficient for the existence of a grazing family of periodic solutions is given the properties of these periodic solutions are discussed a plenty of results on the topological structure of attractors of vibro impact systems is known however since the considered system is strongly nonlinear these attractors must be very sensitive to changes of parameters of the system on the other hand they are observed in experiments and numerical simulations we offer theorem 2 an approach which allows to explain this contradiction and give a new robust mathematical model of the non hyperbolic dynamics in a neighborhood of grazing

arithmetical birational invariants of linear algebraic groups over two dimensional geometric fields let g be a connected linear algebraic group over a geometric field k of cohomological dimension 2 of one of the types which were considered by colliot thelene gille and parimala basing on their results we compute the group of classes of r equivalence g k r the defect of weak approximation a g the first galois cohomology h 1 k g and the tate shafarevich kernel x 1 k g for suitable k in terms of the algebraic fundamental group 1 g we prove that the groups g k r and a g and the set x 1 k g are stably k birational invariants of g

rationally connected varieties over finite fields let x be a geometrically rational or more generally separably rationally connected variety over a finite field k we prove that if k is large enough then x contains many rational curves defined over k as a consequence we prove that r equivalence is trivial on x if k is large enough these results imply the following conjecture of j l colliot thelene let y be a rationally connected variety over a number field f for a prime p let y p denote the corresponding variety over the local field f p then for almost all primes p the chow group of 0 cycles on y p is trivial and r equivalence is trivial on y p

periodic response and stability of oscillators with symmetric trilinear restoring force abstract an exact solution is presented for the periodic symmetric double crossing response of a single degree of freedom strongly non linear oscillator subjected to harmonic excitation the non linearity of the system appears in the restoring force which is a symmetric trilinear function of the system displacement the stability analysis of the periodic solutions is also developed and presented both period one and subharmonic motions are analyzed and investigated for some combinations of the system parameters no stable periodic motion is found to exist and the system exhibits chaotic behavior