Non-asymptotic Regime Research Articles

Discrepancy behaviour in the non-asymptotic regime

We have computed true star-discrepancies D ∗(N) for Halton and Niederreiter point sequences in dimensions s=2, 3, 4 , and 6 with N up to 250 000 , 10 000 , 2000 and 300, respectively. For comparison, we also calculated some L 2 discrepancies T ∗ . The mean behaviour of D ∗(N) can well be approximated by power laws with an exponent between about −0.7 and −0.9, which slowly decreases with N. This behaviour is far from the generally assumed asymptotic one ∼ C( s)(log N) s / N. The factor between the true and the asymptotic behaviour increases strongly with s and reaches many orders of magnitude for large s. The ratios of D ∗(N) for different low discrepancy sequences are not proportional to the presumed asymptotic pre-factors C( s). Especially for the range of bases investigated, Niederreiter sequences have about the same D ∗ as Halton ones in a range of N, which we conjecture to be at least of the order of 10 10.

Performance of space-time codes for a large number of antennas

We study the asymptotic behavior of space-time codes when the number of transmit and receive antennas grows to infinity. Specifically, we determine the behavior of pairwise error probabilities with maximum-likelihood (ML) decoding and with three types of receiver interfaces: the ML interface, the linear zero-forcing (ZF) interface, and the linear minimum-mean-square-error (MMSE) interface. Two situations are studied: when the number of receiving antennas grows to infinity while the number of transmitting antennas is finite, and when both numbers grow to infinity but their ratio remains constant. We show that with ML or linear interfaces the asymptotic performance of space-time codes is determined by the Euclidean distances between codewords. Moreover, with the two linear interfaces examined here the number r of receive antennas must be much larger than the number t of transmit antennas to avoid a sizeable loss of performance; on the other hand, when r /spl Gt/ t, the performance of these linear interfaces comes close to that of ML. The dependence of error probabilities on Euclidean distance is valid for intermediate signal-to-noise ratios (SNRs) even when the number of antennas is small. Simulations validate our theoretical findings, and show how asymptotic results may be substantially valid even in a nonasymptotic regime: thus, even for few antennas, off-the-shelf codes may outperform space-time codes designed ad hoc.

Spin dynamics in Nd 1− xSr xMnO 3 with x=0.5

Muon spin relaxation (μ +SR) spectroscopy has been applied to study spin dynamics in the manganite single crystal Nd 0.5Sr 0.5MnO 3. Critical slowing down of Mn spin fluctuations has been observed above the ferromagnetic ordering temperature T C∼251 K. The dynamic critical exponent z=2.00(12), deduced from muon spin relaxation rates above T C (nonasymptotic regime) agrees with the results of 3d dipolar ferromagnets. Root-exponential relaxation is observed above 160 K and has been attributed to muon spin relaxation in local regions containing differently sized small spin clusters.

New aspects of critical sound attenuation in magnetic systems with spin-lattice relaxation

A general theory of critical sound propagation, including phonon-spin-energy coupling, is studied in anisotropic magnets above their transition temperature. The Kawasaki weak singularity in the ultrasonic attenuation is found as a nonasymptotic effect. A new nonasymptotic regime similar to the one in the binary mixture is also determined. The role of coupling constants and the bare relaxation times in establishing the dominance region of particular terms, is discussed.

Exchange coupling between ferromagnetic layers: Effect of quantum well states (abstract)

It is a well-established experimental result that the exchange coupling between ferromagnetic metals Fe, Co, or Ni is an oscillatory function of the thickness of the paramagnetic spacer. Theories of this coupling are almost in complete unanimity in attributing the observed periodicity to the properties of the Fermi surface of the paramagnetic spacer. Little has been done, and no consensus reached, as to the origin of the strength of the coupling. A realistic assessment of this quantity requires the use of an itinerant picture of the ferromagnetic constituent. From a recent study based on this picture, it became clear that the ratio of the spin-split band gap to the Fermi energy is important in establishing not only the strength of the coupling, but the phase of the oscillations in the nonasymptotic spacer thickness regime. Here we extend the work, which was limited to cases in which the minority-spin band of the ferromagnet was energy matched to the paramagnetic band, to cases in which the match is to that of the majority-spin band. In the latter arrangement, electrons of one spin are trapped in a well whose width equals the thickness of the spacer. We calculated the spin-flip current, and hence, the torque, exerted by one ferromagnet on the other as the magnetization direction of the former is rotated with respect to that of the latter. As in the earlier case, the torque leads to higher-order (in the cosine of the angle between the magnetizations of the ferromagnets) coupling as well as the usual linear coupling. The bound states produced by the quantum well structure are most important when the band splitting in the ferromagnet is large. The new features produced by these states will be discussed; in particular, their effects on the coupling strength and the phase shift will be discussed.

Heat transfer, drag reduction, and fluid characterization for turbulent flow of polymer solutions: recent results and research needs

Some results of investigations on the heat transfer and friction reduction for viscoelastic polymer solutions flowing in circular tubes are presented. The effects of concentration and mechanical degradation on the drag and heat transfer are discussed. Measurements of heat transfer in the thermal entrance region are also reported and indicate a very long entrance region for asymptotic conditions and a shorter one for the non-asymptotic regime. Both these and fully-developed results suggest a departure from the friction/heat transfer analogies, both at the macroscopic coefficient level and at the turbulent diffusivity level. In particular, our computations showed that the turbulent Prandtl number may reach values of up to 15 for asymptotic drag-reducing solutions. In addition to the presentation of these results, the current understanding in this research area is discussed, with a review of some of the most important recent results and of the current research needs in terms of modeling, correlations, experiments, and fluid characterization.

The excluded-volume expansion in polymer chains: Evaluation of the Flory exponent in the Gaussian approximation

The self-consistent excluded-volume chain-expansion problem is reexamined, in the Gaussian approximation. The integral equation resulting from minimization of the free energy is solved analytically on the basis of a conjecture that is shown to be logically consistent. The mean-square distance between atoms separated by k bonds (k≫1) within an infinitely large chain is 〈r2(k)〉∝k6/5{∏∞n=1Ln [ln(k/k̄)]}2/5, where Ln(x) =1+an ln Ln−1(x), L1=1+a1x; k̄ is a short-range cutoff and the ai’s are suitable constants, generally comprised between 0 and 1, while limn→∞ an =1. Accordingly, the divergence associated with the short-range cutoff is explicitly evaluated instead of being removed through renormalization techniques. For (k/k̄)→∞ the above function is independent of k̄ and we have ln〈r2(k)〉∝6/5 ln k, thus reobtaining Flory’s asymptotic exponent. Numerical results obtained from the integral equation in the nonasymptotic regime give support to the above analytic solution.