strong invariance and one sided lipschitz multifunctions abstract this paper studies the strong invariance property of a differential inclusion in finite dimensions under the assumption of a locally one sided lipschitz condition

unramified cohomology and witt groups of anisotropic pfister quadrics the unramified witt group of an anisotropic conic over a field k with char k 0 2 defined by the form 1 a b is known to be a quotient of the witt group w k of k and isomorphic to w k 1 a b ab w k we compute the unramified cohomology group h43rk c where c is the three dimensional anisotropic quadric defined by the quadratic form 1 a b ab c over k we use these computations to study the unramified witt group of c let k be a field of characteristic not two and k k be a finitely generated field extension the n th unramified cohomology group of k over k with coefficients in z 2 cf ct is denoted hnnr k k or just hnnr k when the ground field being considered is clear it is a subgroup of the galois cohomology group hn k z 2 abbreviated to hn k and is defined as hnnr k k n h6nt spec ov u2 vev k here v k is the set of all rank one discrete valuations on k that are trivial on k for v e v k ov denotes the corresponding discrete valuation ring it is well known that the group hnt spec ov j2 injects into hn k if x k is a smooth projective geometrically integral variety and k x is its function field then by abuse of terminology we refer to the groups hnrn k x as the unramified cohomology groups of x for a field f let w f be the witt group of quadratic forms over f sc for k and k k as above we can similarly define the unramified witt group of k k denoted wtr k k as the subgroup of w k defined by wnr k k n w ov vev k here w ov is the witt group of the discrete valuation ring ov which is a subgroup of w k sc chapter vi if x k is smooth projective then we refer to wnr k x as the unramified witt group of x recall that a variety x is said to be k rational if x is k birational to the projective space then the natural maps hn k hn k x resp w k w k x induce isomorphisms htn k hnrr k x resp w k wnr k x thus the unramified cohomology groups of a k rational variety resp the unramified witt received by the editors november 7 1995 1991 mathematics subject classification primary iie70 secondary 13k05 12g05

lyapunov optimization of a damped system our aim is to optimize the damping of a linear vibrating system the penalty function is the average total energy which is equal to the trace of the corresponding lyapunov solution we prove the existence and the uniqueness of the global minimum if the damping varies over the set of all possible positive definite matrices the minimum is shown to be taken on the so called modal critical damping thus confirming a long existing conjecture we also give some preliminary results concerning dampings which depend linearly on the viscosity parameters whereas the damper positions are kept fixed we produce physical examples on which the minimum is taken on a negative viscosity or which have several local minima both phenomena seem to be a consequence of a bad choice of the damper positions

properties of the reachable set of control systems abstract we investigate some properties of the reachable set of a control system representing the system as a differential inclusion and using proximal hamilton jacobi equation we describe its graph we work in infinitely dimensional hilbert space and use one sided lipschitz approach the funnel equation is considered in the last section that equation describes the reachable set in arbitrary banach space we consider also the autonomous case and prove the existence of a limit of the reachable set

motivic spheres and the image of the suslin hurewicz map we show that an old conjecture of a a suslin characterizing the image of a hurewicz map from quillen k theory in degree n to milnor k theory in degree n admits an interpretation in terms of unstable mathbb a 1 homotopy sheaves of the general linear group using this identification we establish suslin s conjecture in degree 5 for infinite fields having characteristic unequal to 2 or 3 we do this by linking the relevant unstable mathbb a 1 homotopy sheaf of the general linear group to the stable mathbb a 1 homotopy of motivic spheres

birational invariants and 1 connectedness we study the a singular chain complex and zeroth a homology sheaf of smooth algebraic varieties over a field k these objects provide quadratic analogs of the motive and zeroth suslin homology sheaf of a smooth scheme in a sense we will explain we show that if k has characteristic 0 the zeroth a homology resp suslin homology sheaf is a birational invariant of smooth proper varieties both are a homotopy invariant we relate these sheaves to various types of unramified invariants in particular unramified cohomology we then show how the techniques in voevodsky s proof of the milnor conjecture and the voevodsky rost proof of its generalization the bloch kato conjecture can be used to study these sheaves using this last set of ideas we provide a construction of unirational non rational varieties where non rationality is detected by non triviality of a degree n unramified etale cohomology group with torsion coefficients yet cannot be detected by lower degree invariants

unramified cohomology of quadrics i given a quadric x over a field f of characteristic 2 we compute the kernel and cokernel of the natural map in degree 4 from the mod 2 galois cohomology of f to the unramified mod 2 cohomology of f x when dim x 10 and in several smaller dimensional cases applications of these results to real quadrics and to the unramified witt ring are given

the arason invariant and mod 2 algebraic cycles introduction 73 1 review of the arason invariant 75 2 the special clifford group 76 3 k cohomology of split reductive algebraic groups 78 4 k cohomology of bg 86 5 gl n and cliff n n 92 6 two invariants for clifford bundles 95 7 snaking a bloch ogus differential 100 8 proof of theorem 1 101 9 application to quadratic forms 102 appendix a toral descent 104 appendix b the rost invariant 108 appendix c an amusing example 113 acknowledgements 116 references 116

damping optimization in mechanical systems with external force abstract we consider a mechanical system excited by external force model of such a system is described by the system of ordinary differential equations m x t d x t kx t f t where matrices m k mass and stiffness are positive definite and the vector f corresponds to an external force the damping matrix d is assumed to be positive semidefinite and has a small rank we introduce two criteria that allow damping optimization of mechanical system excited by an external force since in general a damping optimization is a very demanding problem we provide a new formulas which have been used for efficient damping optimization the efficiency of new formulas is illustrated with a numerical experiment

the graded witt ring and galois cohomology ii a primary problem in the theory of quadratic forms over a field f of characteristic different from two is to prove that the rings hq f and gwf are isomorphic here hqf h gal fq f z 2z where fq is the quadratic closure of f and gwf is the graded witt nrng associated to the fundamental ideal of even dimensional forms in the witt ring wf of f in this paper we assume we are given a field extension k of f such that wk is close to wf or hq k is close to hq f a method is developed to obtain information about these graded rings over f and its 2 extensions from information about the corresponding graded ring of k this relative theory extends and includes the previously developed absolute case where k fq applications are also given to show that hq f and gwf are isomorphic for a collection of fields anrsing naturally from the theory of abstract witt rings

varietes unirationnelles non rationnelles au dela de l exemple d artin et mumford keywords non rational but unirational variety grothendieck invariants rationality brauer invariants reference cma article 1989 001doi 10 1007 bf01850658 record created on 2008 12 16 modified on 2016 08 08

the exponential formula for the reachable set of a lipschitz differential inclusion the main goal of this paper is to prove a formula for the reachable set of a lipschitz differential inclusion with convex values the formula involves a kuratowski limit of sets that resembles a standard approach of defining the exponential of a matrix this explains the title the proof of the main theorem partially relies on a c 1 approximation result due to filippov for which a new proof is given a new approach of characterizing the value function associated with a mayer optimal control problem is given as an application

a uniqueness theorem for differential inclusions abstract the reachable set multifunction of a lipschitz differential inclusion is characterized in terms of a semigroup property and a type of set differentiation at t 0 this generalizes the classical uniqueness theorem of ordinary differential equations a notion of an escape time for differential inclusions is introduced and plays a major role in the main result

a proximal characterization of the reachable set we show that the graph of the reachable set of a control system given by a differential inclusion is uniquely characterized by a hamilton jacobi equation involving proximal normals

birational classification of fields of invariants for groups of order 128 let g be a finite group acting on the rational function field c xg g g c xg g g by cc automorphisms h xg xhgh xg xhg for any g h gg h g noether s problem asks whether the invariant field c g k xg g g gc g k xg g g g is rational i e purely transcendental over cc by fischer s theorem c g c g is rational over cc when g is a finite abelian group saltman and bogomolov respectively showed that for any prime p there exist groups g of order p9p9 and of order p6p6 such that c g c g is not rational over cc by showing the non vanishing of the unramified brauer group brnr c g 0brnr c g 0 which is an avatar of the birational invariant h3 x z torsh3 x z tors given by artin and mumford where x is a smooth projective complex variety whose function field is c g c g for p 2p 2 chu hu kang and prokhorov proved that if g is a 2 group of order 32 then c g c g is rational over cc chu hu kang and kunyavskii showed that if g is of order 64 then c g c g is rational over cc except for the groups g belonging to the two isoclinism families 13 13 with brnr c g 0brnr c g 0 and 16 16 with brnr c g c2brnr c g c2 bogomolov and bohning s theorem claims that if g1g1 and g2g2 belong to the same isoclinism family then c g1 c g1 and c g2 c g2 are stably cc isomorphic we investigate the birational classification of c g c g for groups g of order 128 with brnr c g 0brnr c g 0 moravec showed that there exist exactly 220 groups g of order 128 with brnr c g 0brnr c g 0 forming 11 isoclinism families j j we show that if g1g1 and g2g2 belong to 16 16 31 31 37 37 39 39 43 43 58 58 60 60 or 80 80 resp 106 106 or 114 114 then c g1 c g1 and c g2 c g2 are stably cc isomorphic with brnr c gi c2brnr c gi c2 explicit structures of non rational fields c g c g are given for each cases including also the case 30 30 with brnr c g c2 c2brnr c g c2 c2

fixed set iterations for relaxed lipschitz multimaps abstract a dynamical system described by an autonomous differential inclusion with a right hand side satisfying a relaxed lipschitz condition as well as its euler approximations are studied we investigate the asymptotic properties of the solutions and of the attainable sets it is shown that the system has a strongly flow invariant set or a fixed set that is a set such that each trajectory starting from it does not leave it this set is also an attractor i e it attracts the continuous and the discrete euler trajectories as the time tends to infinity we give estimates of the rate of attraction an algorithm for approximating the fixed set by the attainable sets of the discrete system is also presented

damping optimization over the arbitrary time of the excited mechanical system in this paper we consider damping optimization in mechanical system excited by an external force we use optimization criteria based on minimizing average energy amplitude and average displacement amplitude over the arbitrary time as the main result we derive explicit formulas for objective functions these formulas can be implemented efficiently and accelerate optimization process significantly which is illustrated in a numerical example

measurable viability theorems and the hamilton jacobi bellman equation abstract we prove viability and invariance theorems for systems with dynamics depending on time in a measurable way and having time dependent state constraints x t f t x t x t p t in the above t p t is an absolutely continuous set valued map and t x f t x is a set valued map which is measurable with respect to t and upper semicontinuous or continuous or locally lipschitz with respect to x for this aim we investigate infinitesimal generators of reachable maps and the lebesgue points of set valued maps the results are applied to define and to study lower semicontinuous solutions of the hamilton jacobi bellman equation u t h t x u x 0 with the hamiltonian h measurable with respect to time locally lipschitz with respect to x and convex in the last variable