On ＂Problems on von Neumann Algebras by R. Kadison, 1967＂

A brief summary of the development on Kadison‘s famous problems (1967) is given. A new set of problems in von Neumann algebras is listed.     

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.

     

Unusual Corrections to Scaling in the 3-State Potts Antiferromagnet on a Square Lattice

At zero temperature, the 3-state antiferromagnetic Potts model on a square lattice maps exactly onto a point of the 6-vertex model whose long-distance behavior is equivalent to that of a free scalar boson. We point out that at nonzero temperature there are two distinct types of excitation: vortices, which are relevant with renormalization-group eigenvalue 1/2 and non-vortex unsatisfied bonds, which are strictly marginal and serve only to renormalize the stiffness coefficient of the underlying free boson. Together these excitations lead to an unusual form for the corrections to scaling: for example, the correlation length diverges as J/kT according to Ae ² (1+be ^– +···), where b is a nonuniversal constant that may nevertheless be determined independently. A similar result holds for the staggered susceptibility. These results are shown to be consistent with the anomalous behavior found in the Monte Carlo simulations of Ferreira and Sokal.     

On the Gibbsian Nature of the Random Field Kac Model Under Block-Averaging

We consider the Kac–Ising model in an arbitrary configuration of local magnetic fields = , in any dimension d, at any inverse temperature. We investigate the Gibbs properties of the 'renormalized' infinite volume measures obtained by block averaging any of the Gibbs-measures corresponding to fixed , with block-length small enough compared to the range of the Kac-interaction. We show that these measures are Gibbs measures for the same renormalized interaction potential. This potential depends locally on the field configuration and decays exponentially, uniformly in , for which we give explicit bounds. The construction of the potential is based on a high temperature-type cluster expansion.     

Serre's generalization of Nagao's theorem: An elementary approach

Let be a smooth projective curve over a field . For each closed point of let be the coordinate ring of the affine curve obtained by removing from . Serre has proved that is isomorphic to the fundamental group, , of a graph of groups , where is a tree with at most one non-terminal vertex. Moreover the subgroups of attached to the terminal vertices of are in one-one correspondence with the elements of , the ideal class group of . This extends an earlier result of Nagao for the simplest case .

Serre's proof is based on applying the theory of groups acting on trees to the quotient graph , where is the associated Bruhat-Tits building. To determine he makes extensive use of the theory of vector bundles (of rank 2) over . In this paper we determine using a more elementary approach which involves substantially less algebraic geometry.

The subgroups attached to the edges of are determined (in part) by a set of positive integers , say. In this paper we prove that is bounded, even when Cl is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of , involving unipotent and elementary matrices.

     

Testing multivariate uniformity and its applications

Some new statistics are proposed to test the uniformity of random samples in the multidimensional unit cube These statistics are derived from number-theoretic or quasi-Monte Carlo methods for measuring the discrepancy of points in . Under the null hypothesis that the samples are independent and identically distributed with a uniform distribution in , we obtain some asymptotic properties of the new statistics. By Monte Carlo simulation, it is found that the finite-sample distributions of the new statistics are well approximated by the standard normal distribution, , or the chi-squared distribution, . A power study is performed, and possible applications of the new statistics to testing general multivariate goodness-of-fit problems are discussed.

     

Derived Picard Groups of Finite-Dimensional Hereditary Algebras

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPic _k (A) is the group of triangle auto-equivalences of D> ^b( mod A) induced by two-sided tilting complexes. We study the group DPic _k (A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPic _k , as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPic _k (A) on a certain infinite quiver ^irr. This representation is faithful when the quiver of A is a tree, and then DPic _k (A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic _k (A). When A is hereditary, DPic _k (A) coincides with the full group of k-linear triangle auto-equivalences of D^b( mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to D^b( mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.     

Necessary and sufficient conditions for s-Hopfian manifolds to be codimension-2 fibrators

Fibrators help detect approximate fibrations. A closed, connected -manifold is called a codimension-2 fibrator if each map defined on an -manifold such that all fibre , are shape equivalent to is an approximate fibration. The most natural objects to study are s-Hopfian manifolds. In this note we give some necessary and sufficient conditions for s-Hopfian manifolds to be codimension-2 fibrators.

     

用ABCD矩阵法计算四通式单光栅展宽器的展宽量

以激光束的ABCD传输矩阵理论及Collins积分公式为基础,计算简单实用的四通式单光栅展宽器的色散量,并求出相应展宽量。     

Factorization in finite-codimensional ideals of group algebras

Let G be a -compact, locally compact group and I be a closed2-sided ideal with finite codimension in L¹(G). It is shownthat there are a closed left ideal L having a right boundedapproximate identity and a closed right ideal R having a leftbounded approximate identity such that I = L + R. The proofuses ideas from the theory of boundaries of random walks ongroups. 2000 Mathematics Subject Classification: primary 43A20;secondary 42A85, 43A07, 46H10, 46H40, 60B11.