Unramified cohomology of degree 3 and Noether’s problem

Let G be a finite group and W be a faithful representation of G over C. The group G acts on the field of rational functions C(W). The question whether the field of invariant functions C(W) ^G is purely transcendental over C goes back to Emmy Noether. Using the unramified cohomology group of degree 2 of this field as an invariant, Saltman gave the first examples for which C(W) ^G is not rational over C. Around 1986, Bogomolov gave a formula which expresses this cohomology group in terms of the cohomology of the group G. In this paper, we prove a formula for the prime to 2 part of the unramified cohomology group of degree 3 of C(W) ^G . Specializing to the case where G is a central extension of an F _p -vector space by another, we get a method to construct nontrivial elements in this unramified cohomology group. In this way we get an example of a group G for which the field C(W) ^G is not rational although its unramified cohomology group of degree 2 is trivial. Dedicated to Jean-Louis Colliot-Thélène.     

Multiplication operators on sobolev disk algebra

In this paper,we study the algebra consisting of analytic functions in the Sobolev space W~(2,2) (D) (D is the unit disk),called the Sobolev disk algebra,explore the properties of the multiplication operators M_f on it and give the characterization of the corn- mutant algebra A′(M_f) of M_f.We show that A′(M_f) is commutative if and only if M_f~* is a Cowen-Douglas operator of index 1.     

Dynamics of polynomial automorphisms ofC k

We study the dynamics of polynomial automorphisms ofC ^k . To an algebraically stable automorphism we associate positive closed currents which are invariant underf, consideringf as a rational map onP ^k . These currents give information on the dynamics and allow us to construct a canonical invariant measure which is shown to be mixing.     

On the Convergence in Capacity of Rational Approximants

Let {r _n } be a sequence of rational functions deg( r _n ≤ n) that converge rapidly in measure to an analytic function f on an open set in C ^N . We show that {r _n } converges rapidly in capacity to f on its natural domain of definition W _f (which, by a result of Goncar, is an open subset of C ^N ). In particular, for f meromorphic on C ^N and analytic near zero the sequence of Padé approximants {π _n (z, f, λ)} (as defined by Goncar) converges rapidly in capacity to f on C ^N . January 14, 1999. Date revised: October 7, 1999. Date accepted: November 1, 1999.     

Approximate Factorization in Shallow Water Applications

We consider the numerical integration of problems modelling phenomena in shallow water in 3 spatial dimensions. If the governing partial differential equations for such problems are spatially discretized, then the right-hand side of the resulting system of ordinary differential equations can be split into terms f ₁, f ₂, f ₃ and f ₄, respectively representing the spatial derivative terms with respect to the x, y and z directions, and the interaction terms. It is typical for shallow water applications that the interaction term f ₄ is nonstiff and that the function f ₃ corresponding with the vertical spatial direction is much more stiff than the functions f ₁ and f ₂ corresponding with the horizontal spatial directions. The reason is that in shallow seas the gridsize in the vertical direction is several orders of magnitude smaller than in the horizontal directions. In order to solve the initial value problem (IVP) for these systems numerically, we need a stiff IVP solver, which is necessarily implicit, requiring the iterative solution of large systems of implicit relations. The aim of this paper is the design of an efficient iteration process based on approximate factorization. Stability properties of the resulting integration method are compared with those of a number of integration methods from the literature. Finally, a performance test on a shallow water transport problem is reported.     

Linearization of functions

Given a space (U) of functions f:U which are continuous, we construct another space _*(U) and a map e:U *(U) linearizing all functions f (U) (i.e. there are L_{f *}(U) such that L_f^e=f). Such linearizations are stronger than mere preduals for (U), for example for (U)=₁, linearizations correspond to preduals of ₁ which are isomorphic to c₀. We also address the vector-valued case. A number of such linearizing constructions are to be found in the literarture, mostly for certain spaces of holomorphic functions. The procedure presented here generalizes all these special cases.Mathematics Subject Classification (2000):46E10, 46G20     

Classroom note: Using Platonic bodies to generate a random sample from a population having two characteristics linked with a predetermined coefficient of correlation

This note describes a simple method of generating a random sample of N pairs (U _i ,?W _i ) from a population whose elements have two characteristics U  and W associated with a known coefficient of correlation. Although the method described is extremely advantageous when Platonic bodies are used, it can be generalized to include any discrete uniform distribution.     

Duality and multiplication operators

We describe a space of functions contained inxxLx (D)C(D G) but not necessarily inU. We give a representation of these functions as bounded multiplication operators on the Bergman spacexxLx _a ² and identify the subspace consisting of functions which induce compact multiplication operators. We also describe a newC ^*-subalgebra ofxxLx (D) which we conjecture to be a proper super-set ofU.Most of this research was done while the second author was visiting Cleveland State University. He would like to thank the Mathematics Department for its hospitality. He would also like to thank the NNSFC for partial support.     

Partial Parity (g,f )-Factors and Subgraphs Covering Given Vertex Subsets

 Let G be a graph and W a subset of V(G). Let g,f:V(G)→Z be two integer-valued functions such that g(x)≤f(x) for all x∈V(G) and g(y)≡f(y) (mod 2) for all y∈W. Then a spanning subgraph F of G is called a partial parity (g,f)-factor with respect to W if g(x)≤deg _F (x)≤f(x) for all x∈V(G) and deg _F (y)≡f(y) (mod 2) for all y∈W. We obtain a criterion for a graph G to have a partial parity (g,f)-factor with respect to W. Furthermore, by making use of this criterion, we give some necessary and sufficient conditions for a graph G to have a subgraph which covers W and has a certain given property. Received: June 14, 1999?Final version received: August 21, 2000     

Algebraically constructible functions and signs of polynomials

LetW be a real algebraic set. We show that the following families of integer-valued functions onW coincide: (i) the functions of the formω →λ(X _ω ), where X _ω are the fibres of a regular morphismf :X →W of real algebraic sets, (ii) the functions of the formω →χ(X _ω ), where X _ω are the fibres of a proper regular morphismf :X →W of real algebraic sets, (iii) the finite sums of signs of polynomials onW. Such functions are called algebraically constructible onW. Using their characterization in terms of signs of polynomials we present new proofs of their basic functorial properties with respect to the link operator and specialization. Research partially supported by an Australian Research Council Small Grant. Second author also partially supported by KBN 610/P3/94.     

The number of connected sparsely edged graphs. II. Smooth graphs and blocks

A smooth graph is a connected graph without endpoints; f(n, q) is the number of connected graphs, v(n, q) is the number of smooth graphs, and u(n, q) is the number of blocks on n labeled points and q edges: W_k, V_k, and U_k are the exponential generating functions of f(n, n + k), v(n, n + k), and u(n, n + k), respectively. For any k ? 1, our reduction method shows that V_k can be deduced at once from W_k, which was found for successive k by the computer method described in our previous paper. Again the reduction method shows that U_k must be a sum of powers (mostly negative) of 1 - X and, given this information, we develop a recurrence method well suited to calculate U_k for successive k. Exact formulas for v(n, n + k) and u(n, n + k) for general n follow at once.     

On One Problem of Group Classification of Linear Dynamical Systems

The set of all linear control systems over the field of complex numbers C with m=dim _C U inputs and n=dim _C X states is studied from the point of view of algebraic geometry. We give a classification of this set with respect to a simultaneous change of bases of the spaces X and U. We also find a subset of all pre-stable systems. In the case where m=2 we construct the factors of principal open sets covering W (the polynomial f is a relative invariant). Assuming the same conditions to hold, we find generators of the ring of invariants. These results are also partially generalized to the case of systems with outputs. We show an application of the theory of invariants to the problem of parametric identification of systems.     

Lie algebra of formal vector fields which are extended by formal g-valued functions

In this paper, we consider the infinite-dimensional Lie algebra W_n⋉g⊗O _n of formal vector fields on the n-dimensional plane which is extended by formal g-valued functions of n variables. Here g is an arbitrary Lie algebra. We show that the cochain complex of this Lie algebra is quasi-isomorphic to the quotient of the Weyl algebra of (gl _n ⊕ g) by the (2n+1)st term of the standard filtration. We consider separately the case of a reductive Lie algebra g. We show how one can use the methods of formal geometry to construct characteristic classes of bundles. For every G-bundle on an n-dimensional complex manifold, we construct a natural homomorphism from the ring A of relative cohomologies of the Lie algebra W_n⋉g⊗O _n to the ring of cohomologies of the manifold. We show that generators of the ring A are mapped under this homomorphism to characteristic classes of tangent and G-bundles. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 205–230.     

Clairaut relation for geodesics of Hopf tubes

Summary In this note we use the Hopf map π: S³ → S² to construct an interesting family of Riemannian metrics h^fon the 3-sphere, which are parametrized on the space of positive smooth functions f on the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. The Hopf tube over an immersed curve γ in S² is the complete lift π^-1(γ) of γ, and we prove that any geodesic of this Hopf tube satisfies a Clairaut relation. A Hopf tube plays the role in S³ of the surfaces of revolution in R^3. Furthermore, we show a geometric integration method of the Frenet equations for curves in those non-standard S³. Finally, if we consider the sphere S³ equipped with a family h^f of Lorentzian metrics, then a new Clairautrelation is also obtained for timelike geodesics of the Lorentzian Hopf tube, and a geometric integration method for curves is still possible.     

Sobolev functions whose inner trace at the boundary is zero

Let Ω⊂R ⁿ be an arbitrary open set. In this paper it is shown that if a Sobolev functionf∈W ^1,p (Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, thenf is weakly zero on ϖΩ in the sense thatf∈W ₀ ^1,p (Ω).     

Stability of Solution of a Class of Quasilinear Elliptic Equations

In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equationsin space W(θ,p)(Ω), which is bigger than W1,p(Ω). Next, using revise reverse Holder inequality we prove that if ωc is uniformly p-think, then there exists a neighborhood U of p, such that for all t ∈U, the weak solutions of equation corresponding t are bounded uniformly. Finally, we get the stability of weak solutions on exponent p.     

A function space model approach to the rational evaluation subgroups

Let f : U → X be a map from a connected nilpotent space U to a connected rational space X. The evaluation subgroup G _*(U, X; f), which is a generalization of the Gottlieb group of X, is investigated. The key device for the study is an explicit Sullivan model for the connected component containing f of the function space of maps from U to X, which is derived from the general theory of such a model due to Brown and Szczarba (Trans Am Math Soc 349, 4931–4951, 1997). In particular, we show that non Gottlieb elements are detected by analyzing a Sullivan model for the map f and by looking at non-triviality of higher order Whitehead products in the homotopy group of X. The Gottlieb triviality of a fibration in the sense of Lupton and Smith (The evaluation subgroup of a fibre inclusion, 2006) is also discussed from the function space model point of view. Moreover, we proceed to consideration of the evaluation subgroup of the fundamental group of a nilpotent space. In consequence, the first Gottlieb group of the total space of each S ¹-bundle over the n-dimensional torus is determined explicitly in the non-rational case.      

On Lattice Structure of the Space of Pointwise Limits of Harmonic Functions

Let H¹(U) denote the space of all pointwise limits of bounded sequences from H(U), where H(U) consists of all continuous functions on the closure [`(U)]\overline{U} of a bounded open set U⊂ℝ^m that are harmonic on U. It is shown that the space H¹(U) is a lattice in the natural ordering if and only if the set ∂_regU of all regular points of U is an F_σ-set.     

Higher Power Residue Codes

For certain primeslandp, and characters χ:F*_p→F*_l², we construct codesWof lengthp+ 1 overF_l²which are linear overF_l, but not overF_l², and which are invariant under a monomial action of the group SL(2,p). We consider the cases of cubic and quartic characters in detail and use theWto construct linear codes overF_lin these cases.