The diophantine equationx 2+2 k =y n

The author wishes to thank the referee most sincerely for the observation immediately following the statement of the theorem.     

A ratio limit theorem for the finite nonhomogeneous Markov chains

A new simple proof concerning the structure of the tail σ-field of a nonhomogeneous Markov chain as well as a result regarding the asymptotic behaviour of the ratio between the transition probabilities and the absolute probabilities of the chain are given. It is shown that the latter result can be used to characterize the atomic sets of the tail σ-field.     

Economical triangle-square dissection

In the 1830's, it was proved almost simultaneously by W. Bolyai and Gerwien that given any two polygons of equal area, it is possible to dissect one into a finite number of pieces which may be rearranged to form the other. In this paper the dissection of a triangle to give a square is considered. No estimate has been made previously on the economy of such a dissection. A dissection which is probably close in economy to the most economical dissection is found, and an estimate is made as to how uneconomical any dissection must be.     

On the class number of certain imaginary quadratic fields

Theorem. Let 2$"> denote an integer, the square-free part of and the class number of the field . Then except for the case , divides .

     

Best Constants for Moser-Trudinger Inequalities, Fundamental Solutions and One-Parameter Representation Formulas on Groups of Heisenberg Type

We derive the explicit fundamental solutions for a class of degenerate (or singular) one-parameter subelliptic differential operators on groups of Heisenberg (H) type. This extends the results of Kaplan of the sub-Laplacian on H-type groups, which in turn generalizes Folland's result on the Heisenberg group. As an application, we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups. By choosing the parameter equal to the homogeneous dimension Q and using the Moser-Trudinger inequality for the convolutional type operator on stratified groups obtained in [18], we get the following theorem which gives the best constant for the Moser-Trudinger inequality for Sobolev functions in H-type groups. Let ${\Bbb G}We derive the explicit fundamental solutions for a class of degenerate (or singular) one-parameter subelliptic differential operators on groups of Heisenberg (H) type. This extends the results of Kaplan of the sub-Laplacian on H-type groups, which in turn generalizes Folland's result on the Heisenberg group. As an application, we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups. By choosing the parameter equal to the homogeneous dimension Q and using the Moser-Trudinger inequality for the convolutional type operator on stratified groups obtained in [18], we get the following theorem which gives the best constant for the Moser-Trudinger inequality for Sobolev functions in H-type groups. Let ? be any group of Heisenberg type whose Lie algebra is g enerated by m left invariant vector fields and with a q-dimensional center. Let and Then, with A _Q as the sharp constant, where ∇_? denotes the subellitpic gradient on ? This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group [18]. Received March 15, 2001, Accepted September 21, 2001     

First symmetry tests in polarized Z0 decays to bbg

We have made the first direct symmetry tests in the decays of polarized Z0 bosons into fully identified bbg states, collected in the SLD experiment at SLAC. We searched for evidence of parity violation at the bbg vertex by studying the asymmetries in the b-quark polar- and azimuthal-angle distributions, and for evidence of T-odd, CP-even or CP-odd, final-state interactions by measuring angular correlations between the three-jet plane and the Z0 polarization. We found results consistent with standard model expectations and set 95% C. limits on anomalous contributions.