reduction in the rationality problem for multiplicative invariant fields abstract for a faithful zg lattice a and a field k on which the group g acts by field automorphisms let r be the normal subgroup generated by the elements of g which act trivially on k and act as reflections on a we prove that the rationality of the multiplicative invariant field k a g over k ar g is equivalent to the rationality of k a g over k ar g where g is a particular subgroup of g such that g r g we then use this reduction result to prove that k a g is rational over k where g is the automorphism group of a crystallographic root system g acts trivially on k and a is any lattice on the space q

the covariant form of the klein kramers equation and the associated moment equations we provide a covariant coordinate free formulation of the many dimensional klein kramers equation for the phase space distribution of a brownian particle we construct a complete set of eigenfunctions of the collision operator adapted to the coordinate system which involve covariant tensorial hermite polynomials the klein kramers equation can then be reformulated as a system of coupled equations for the expansion coefficients with respect to this system truncation of this system of moment equations and application of a subsidiary condition yields a covariant generalization of grad s thirteen moment equations as an application we give the explicit form of these equations for spherically symmetric stationary solutions in spherical coordinates we briefly comment on possible extensions of our treatment to slightly more complicated cases

twisted multiplicative field invariants noether s problem and galois extensions noether s problem concerns fields f xg 1 g e g g f g where f is a field g is a finite group g acts by permuting the indeterminants x in the obvious way and f g refers to the field of invariant elements in showing the f g s were nonrational for certain g the author was led to consider multiplicative invariant fields ss which are defined as follows let g be a finite group and q a g lattice that is q is a finitely generated z g module which is free as an abelian group form the group algebra f g and its field of fractions f q g acts naturally on f q and the multiplicative invariant field is f q in ss it was shown that for certain q f q g was essentially equivalent to actually stably isomorphic to f g for g a split extension of g with abelian kernel this leads one to ask whether f g can be expressed as a kind of invariant field for g when g is a nonsplit extension of g the answer is yes and is the core of this paper in fact f g will be shown to be a so called a twisted multiplicative invariant field of g the goal of this paper is to present this fact and then draw a series of conclusions using it such a twisted invariant fields have appeared explicitly in previous works in ss these fields arise as another way of describing the invariant fields of reductive algebraic groups note that in s6 the finite group is the weyl group and the lattice is derived from the root lattice in s7 the unramified brauer groups of a twisted invariant fields are calculated

gersten s conjecture for thek theory with torsion coefficients of a discrete valuation ring between quillen k groups induced by the obvious inclusion functors are all zero in 6 quillen proved this conjecture if r is essentially of finite type over a field in the case of discrete valuation rings this conjecture reduces to conjecture let r be a discrete valuation ring with residue field k and field of fractions f then for all q z 0 the transfer map which is defined since every k vector space is of finite projective dimension as an r module k k k r is zero equivalently the map r ec f induced by the ring homomorphism r p f is injective a third equivalent formulation is that the boundary map

symmetric functions and the phase problem in crystallography the calculation of crystal structure from x ray diffraction data requires that the phases of the structure factors fourier coefficients determined by scattering be deduced from the absolute values of those structure factors motivated by a question of herbert hauptman we consider the problem of determining phases by direct algebraic means in the case of crystal structures with n equal atoms in the unit cell with n small we rephrase the problem as a question about multiplicative invariants for a particular finite group action we show that the absolute values form a generating set for the field of invariants of this action and consider the problem of making this theorem constructive and practical the most promising approach for deriving explicit formulas uses sagbi bases

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

comments on the grad procedure for the fokker planck equation if the kinetic equation of a macroscopic system is expanded with respect to the velocity in terms of orthogonal functions e g in terms of hermite functions one obtains an infinite hierarchy of equations for the expansion coefficients grad s method consists in truncating this hierarchy and investigating the remaining finite system in this paper we set up conditions under which this procedure is rigorously justified in case of the fokker planck equation

the systematic adiabatic elimination of fast variables from a many dimensional fokker planck equation the chapman enskog method for the adiabatic elimination of fast variables is applied to a general fokker planck equation linear in the fast variables this equation is the counterpart of the generalized haken zwanzig model a system of coupled langevin equations often encountered in quantum optics and in the theory of non equilibrium phase transitions after a few equilibration times for the fast variables the time dependence of smooth solutions of this fokker planck equation is completely governed by the reduced distribution function of the slow variables which obeys a closed evolution equation we obtain an explicit perturbation series for the generator of this reduced evolution the system considered here is the most general system for which the chapman enskog hierarchy can be solved explicitly by exploiting an analogy between the unperturbed operator and the hamiltonian for coupled harmonic oscillators

zero cycles on rational surfaces over number fields let x be a smooth projective geometrically integral surface defined over a perfect field k let kbe an algebraic closure of k and f be the function field of x k x then x is said to be rational if f is a purely transcendental extension of k the tame symbols define a map from ke f to the sum of function fields h k 7 of all irreducible divisors 7 on x let m be the cokernel of this map if k is a number field and kv the completion at a place v of k let kv be an algebraic closure of kv and m the corresponding cokernel for k x x further let i i i l k m be the kernel of the natural map from hl gal k k m to i h1 gal k v kv my in 5 w colliot th616ne and sansuc conjectured that

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

rationality problem for generic tori in simple groups abstract we prove that except for several already known cases the generic torus of a simple adjoint or simply connected group is not stably rational this confirms a conjecture by le bruyn on generic norm tori

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

the kinetic boundary layer for the klein kramers equation a new numerical approach we explore a numerical technique for determining the structure of the kinetic boundary layer of the klein kramers equation for noninteracting brownian particles in a fluid near a wall that absorbs the brownian particles the equation is of interest in the theory of diffusion controlled reactions and of the coagulation of colloidal suspensions by numerical simulation of the langevin equation equivalent to the klein kramers equation we amass statistics of the velocities at the first return to the wall and of the return times for particles injected into the fluid at the wall with given velocities the data can be used to construct the solutions of the standard problems at an absorbing wall the milne and the albedo problem we confirm and extend earlier results by burschka and titulaer obtained by a variational method vexed by the slow convergence of the underlying eigenfunction expansion we briefly discuss some further boundary layer problems that can be attacked by exploiting the results reported here

the kinetic boundary layer for the fokker planck equation with absorbing boundary the fokker planck equation for the distribution of position and velocity of a brownian particle is a particularly simple linear transport equation its normal solutions and an apparently complete set of stationary boundary layer solutions can be determined explicitly by a numerical algorithm we select linear combinations of them that approximately fulfill the boundary condition for a completely absorbing plane wall and that approach a linearly increasing position space density far from the wall various aspects of these approximate solutions are discussed in particular we find that the extrapolated asymptotic density reaches zero at a distance xm beyond the wall we find xm 1 46 in units of the velocity persistence length of the brownian particle this study was motivated by certain problems in the theory of diffusion controlled reactions and the results might be used to test approximate theories employed in that field