Reducibility and nonreducibility between equivalence relations

We show that, for , the relation of -equivalence between infinite sequences of real numbers is Borel reducible to the relation of -equivalence (i.e., the Borel cardinality of the quotient is no larger than that of ), but not vice versa. The Borel reduction is constructed using variants of the triadic Koch snowflake curve; the nonreducibility in the other direction is proved by taking a putative Borel reduction, refining it to a reduction map that is not only continuous but `modular,' and using this nicer map to derive a contradiction.

     

On the Diversity Order of Spatial Multiplexing Systems With Transmit Antenna Selection: A Geometrical Approach

In recent years, the remarkable ability of multiple-input-multiple-output (MIMO) wireless communication systems to provide spatial diversity or multiplexing gains has been clearly demonstrated. For MIMO diversity schemes, it is well known that antenna selection methods that optimize the postprocessing signal-to-noise ratio (SNR) can preserve the diversity order of the original full-size MIMO system. On the other hand, the diversity order achieved by antenna selection in spatial multiplexing systems, especially those exploiting practical coding and decoding schemes, has not thus far been rigorously analyzed. In this paper, a geometrical framework is proposed to theoretically analyze the diversity order achieved by transmit antenna selection for separately encoded spatial multiplexing systems with linear and decision-feedback receivers. When two antennas are selected from the transmitter, the exact achievable diversity order is rigorously derived, which previously only appears as conjectures based on numerical results in the literature. If more than two antennas are selected, we give lower and upper bounds on the achievable diversity order. Furthermore, the same geometrical approach is used to evaluate the diversity-multiplexing tradeoff in spatial multiplexing systems with transmit antenna selection     

On the error probability of signals in additive white Gaussiannoise

A new upper bound to the probability of error in detecting one of M equally probable signals in additive white Gaussian noise is presented. This bound is easy to calculate, can be applied to any signal set. It is always better than the union and minimum distance bounds. Examples demonstrate the use of the bound     

Inhibierung der Autoxydation von Polypropylenoxid durch KOH

Ohne Zusammenfassung     

The dynamic stereochemistry of the allylic ligand in a μ-allylic complex of platinum(II)

¹H NMR studies of an equilibrium mixture of 1,1,1,5,5,5-hexafluoropentane-2,4-dionato-π-allylplatinum(II) [IA] and bis(1,1,1,5,5,5-hexafluoropentane-2,4-dionato)bis(μ-allyl)diplatinum(II) [IB] show that the μ-allylic ligands of (IB) exhibit temperature dependent tautomeric behaviour, giving rise to a site-exchange between protons on these ligands.     

Transition-matrix construction for pseudorandom binary-sequence generators

A method is described for rapidly finding the elements of a matrix operator which shifts the state of a class of linear sequence generators through a specified time interval. The method is applied to the problem of generating pseudorandom binary sequences with specified time shifts.     

Quaternary adamantine selenides and tellurides of the form I III IV VI4

The preparation of a range of polycrystalline Selenides and Tellurides of the form I III IV VI₄ was undertaken. Twelve Selenides with quaternary composition were prepared but the eight Tellurides attempted did not grow as quaternaries. Single crystal samples of CuGaSnSe₄ were grown by vapour transport. X-Ray powder photography confirmed that the quaternary Selenides have tetragonal Chalcopyrite structure with a c/a ratio of approximately 2. A systematic variation of unit cell volume and of c/a was found over the entire range of compounds grown.     

Vibronic properties of the F+ centre in strontium oxide

The shape and moments of the F⁺ absorption band in SrO have been measured between 4.2K and 300K. Analysis of the moments gives the effective frequency of the vibrational modes interacting with the centre as $\text{h}\text{?}\text{ω = 236 ± 20 cm}{}^{-1}$, and the Huang-Rhys factors for cubic and non-cubic modes as S_C = 3 ± 1 and S_NC = 4 ± 1 respectively.