Space structure vibration modes: How many exist? Which ones are important?

This report attempts to shed some light on the two issues raised in the title, namely, how many vibration modes does a real structure have, and which of these modes are important? The surprise-free answers to these two questions are, respectively, an infinite number and the first several modes. The author argues that the absurd subspace (all but the first billion modes) is not a strength of continuum modeling, but, in fact, a weakness. Partial differential equations are not real structures, only mathematical models. This note also explains (1) that the PDE model and the finite element model are, in fact, the same model, the latter being a numerical method for dealing with the former, (2) that modes may be selected on dynamical grounds other than frequency alone, and (3) that long slender rods are useful as primitive cases but dangerous to extrapolate from

Linear statistics for zeros of Riemann's zeta function

We consider a smooth counting function of the scaled zeros of the Riemann zeta function, around height T. We show that the first few moments tend to the Gaussian moments, with the exact number depending on the statistic considered

The zeros of random polynomials cluster uniformly near the unit circle

In this paper we deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erdos and Turan. More precisely, given a sequence of random polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, and their angles are uniformly distributed. The method we use is deterministic: in particular, we do not assume independence or equidistribution of the coefficients of the polynomial.Comment: Corrects some typos and strengthens Theorem

The color of sea level: importance of spatial variations in spectral shape for assessing the significance of trends

We investigate spatial variations in the shape of the spectrum of sea level variability, based on a homogeneously-sampled 12-year gridded altimeter dataset. We present a method of plotting spectral information as color, focusing on periods between 2 and 24 weeks, which shows that significant spatial variations in the spectral shape exist, and contain useful dynamical information. Using the Bayesian Information Criterion, we determine that, typically, a 5th order autoregressive model is needed to capture the structure in the spectrum. Using this model, we show that statistical errors in fitted local trends range between less than 1 and more than 5 times what would be calculated assuming “white” noise, and the time needed to detect a 1 mm/yr trend ranges between about 5 years and many decades. For global-mean sea level, the statistical error reduces to 0.1 mm/yr over 12 years, with only 2 years needed to detect a 1 mm/yr trend. We find significant regional differences in trend from the global mean. The patterns of these regional differences are indicative of a sea level trend dominated by dynamical ocean processes, over this perio

Mock-Gaussian Behaviour for Linear Statistics of Classical Compact Groups

We consider the scaling limit of linear statistics for eigenphases of a matrix taken from one of the classical compact groups. We compute their moments and find that the first few moments are Gaussian, whereas the limiting distribution is not. The precise number of Gaussian moments depends upon the particular statistic considered

TACAN operational description for the space shuttle orbital flight test program

The TACAN subsystems (three TACAN transponders, six antennas, a subsystem operating program, and redundancy management software in a tutorial form) are discussed and the interaction between these subsystems and the shuttle navigation system are identified. The use of TACAN during the first space transportation system (STS-1), is followed by a brief functional description of the TACAN hardware, then proceeds to cover the software units with a view to the STS-1, and ends with a discussion on the shuttle usage of the TACAN data and anticipated performance