Differential Simplicity in Polynomial Rings and Algebraic Independence of Power Series

Let k be a field of characteristic zero, f(X,Y), g(X,Y)k[X,Y],g(X,Y) (X,Y) and d:=g(X,Y)/X + f(X,Y)/Y. A connection is establishedbetween the d-simplicity of the local ring k[X,Y]_(X,Y) and thetranscendency of the solution in tk[[t]] of the algebraic differentialequation g(t,y(t))·(/t)y(t)+f(t,y(t)). This connectionis used to obtain some interesting results in the theory ofthe formal power series and to construct new examples of differentiallysimple rings.     

Finite Groups of Bounded Exponent

By means of a simplification in a significant step of Zelmanov'ssolution of the restricted Burnside problem, the known boundsfor the order of finite m-generator groups of exponent n areimproved.     

On the product system of a completely positive semigroup

Given a W∗-continuous semigroup φ of unital, normal, completely positive maps of B(H), we introduce its continuous tensor product system . If α is a minimal dilation E₀-semigroup of φ with Arveson product system F, then is canonically isomorphic to F. We apply this construction to a class of semigroups of arising from a modified Weyl-Moyal quantization of convolution semigroups of Borel probability measures on . This class includes the heat flow on the CCR algebra studied recently by Arveson. We prove that the minimal dilations of all such semigroups are completely spatial, and additionally, we prove that the minimal dilation of the heat flow is cocyle conjugate to the CAR/CCR flow of index two.     

A note on triangular derivations of

For a field of characteristic zero, and for each integer , we construct a triangular derivation of whose ring of constants, though finitely generated over , cannot be generated by fewer than elements.

     

On Non-Equivalence of Quadratic Forms on a Riemannian Manifold

An example is constructed of a manifold for which two quadraticforms corresponding to the fourth-order invariant operatorsare not equivalent. 2000 Mathematics Subject Classification58C40.     

Ascending Hnn Extensions of Finitely Generated Free Groups Are Hopfian

It is shown that every ascending HNN extension of a finitelygenerated free group is Hopfian. An important ingredient inthe proof is that under certain hypotheses on the group H, ifG is an ascending HNN extension of H, then cd(G) = cd(H) + 1.     

Shifted identical bands: A new phenomenon

The levels in ¹⁶²Gd were identified in spontaneous fission studies. Its transition energies are remarkably similar to those in ¹⁶⁰Gd. From that work, an analysis of yrast bands in even-even proton to neutron-rich Ba to Pb nuclei led to the discovery of a new phenomenon, shifted identical bands (SIB). SIBs are yrast bands in neighboring nuclei (a, b) with moments of inertia which are identical when shifted by a constant amount κ, so J _1a (1+κ)=J _1b , from 2⁺ to 8⁺ and higher to 16⁺. Out of over 700 comparisons, 55 SIBs were found from stable to the most neutron-rich Ce-W nuclei with $\left| {\bar k} \right|$ between 1.5% and 13%, where the spread in κ is less than ±1%, and only four identical bands ( $\bar k \cong 0$ ). As examples, we found for ¹⁵⁸Sm-¹⁶⁰Gd, $\bar k = \left( { - 3.2_{ - 0.2}^{ + 0.1} } \right)\%$ (where the ± is the total spread in κ from ?3.1 to ?3.4); ¹⁵⁶Nd-¹⁶⁰Gd, (?10.6 _?0.2 ^+0.4 )%; ¹⁵⁸Sm-¹⁶⁰Sm, (3.4 _?0.3 ^+0.5 )%. The J ₁ values were fitted to a variable moment of inertia model with parameters J ₀ and C whose values correlate with the SIB J ₁ values. The SIBs are not correlated either with deformation or with the N _p N _n product of the IBA model.     

Bounded composition operators with closed range on the Dirichlet space

For composition operators on spaces of analytic functions it is well known that norm estimates can be converted to Carleson measure estimates. The boundedness of the composition operator becomes equivalent to a Carleson measure inequality. The measure corresponding to a composition operator on the Dirichet space is , where is the cardinality of the preimage . The composition operator will have closed range if and only if the corresponding measure satisfies a ``reverse Carleson measure' theorem: for all . Assuming is bounded, a necessary condition for this inequality is a reverse of the Carleson condition: (C) for all Carleson squares . It has long been known that this is not sufficient for a completely general measure. Here we show that it is also not sufficient for the special measures . That is, we construct a function such that is bounded and satisfies (C) but the composition operator does not have closed range.

     

Hyperbolic minimizing geodesics

We apply the intersection theory for Lagrangian submanifolds to obtain a Sturm type comparison theorem for linearized Hamiltonian flows. Applications to the theory of geodesics are considered, including a sufficient condition that arclength minimizing closed geodesics, for an -dimensional Riemannian manifold, are hyperbolic under the geodesic flow. This partially answers a conjecture of G. D. Birkhoff.