The true complexity of a system of linear equations

It is well-known that if a subset A of a finite Abelian group G satisfies a quasirandomness property called uniformity of degree k, then it contains roughly the expected number of arithmetic progressions of length k, that is, the number of progressions one would expect in a random subset of G of the same density as A. One is naturally led to ask which degree of uniformity is required of A in order to control the number of solutions to a general system of linear equations. Using so-called "quadratic Fourier analysis", we show that certain linear systems that were previously thought to require quadratic uniformity are in fact governed by linear uniformity. More generally, we conjecture a necessary and sufficient condition on a linear system L which guarantees that any subset A of F_p^n which is uniform of degree k contains the expected number of solutions to L.Comment: 30 page

Linear forms and higher-degree uniformity for functions on Fpn\mathbb{F}_p^n

In [GW09a] we conjectured that uniformity of degree $k-1$ is sufficient to control an average over a family of linear forms if and only if the $k$th powers of these linear forms are linearly independent. In this paper we prove this conjecture in $\mathbb{F}_p^n$, provided only that $p$ is sufficiently large. This result represents one of the first applications of the recent inverse theorem for the $U^k$ norm over $\mathbb{F}_p^n$ by Bergelson, Tao and Ziegler [BTZ09,TZ08]. We combine this result with some abstract arguments in order to prove that a bounded function can be expressed as a sum of polynomial phases and a part that is small in the appropriate uniformity norm. The precise form of this decomposition theorem is critical to our proof, and the theorem itself may be of independent interest.Comment: 40 page

Generalized Miller Formulae

We derive the spectral dependence of the non-linear susceptibility of any order, generalizing the common form of Sellmeier equations. This dependence is fully defined by the knowledge of the linear dispersion of the medium. This finding generalizes the Miller formula to any order of non-linearity. In the frequency-degenerate case, it yields the spectral dependence of non-linear refractive indices of arbitrary order.Comment: 12 pages, 1 figure (4 panels

Investigation of line-of-sight propagation in dense atmosphere, phase 2 Final report, Jun. 1970 - Feb. 1971

Effect of microwave absorption and decimetric radio noise in Jovian atmospheres on radio communication in 1 to 10 GHz frequency ban

Self streamlining wind tunnel: Low speed testing and transonic test section design

Comprehensive aerodynamic data on an airfoil section were obtained through a wide range of angles of attack, both stalled and unstalled. Data were gathered using a self streamlining wind tunnel and were compared to results obtained on the same section in a conventional wind tunnel. The reduction of wall interference through streamline was demonstrated

Highlights of experience with a flexible walled test section in the NASA Langley 0.3-meter transonic cryogenic tunnel

The unique combination of adaptive wall technology with a contonuous flow cryogenic wind tunnel is described. This powerful combination allows wind tunnel users to carry out 2-D tests at flight Reynolds numbers with wall interference essentially eliminated. Validation testing was conducted to support this claim using well tested symmetrical and cambered airfoils at transonic speeds and high Reynolds numbers. The test section hardware has four solid walls, with the floor and ceiling flexible. The method of adapting/shaping the floor and ceiling to eliminate top and bottom wall interference at its source is outlined. Data comparisons for different size models tested and others in several sophisticated 2-D wind tunnels are made. In addition, the effects of Reynolds number, testing at high lift with associated large flexible wall movements, the uniqueness of the adapted wall shapes, and the effects of sidewall boundary layer control are examined. The 0.3-m TCT is now the most advanced 2-D research facility anywhere

Predictive wall adjustment strategy for two-dimensional flexible walled adaptive wind tunnel: A detailed description of the first one-step method

Following the realization that a simple iterative strategy for bringing the flexible walls of two-dimensional test sections to streamline contours was too slow for practical use, Judd proposed, developed, and placed into service what was the first Predictive Strategy. The Predictive Strategy reduced by 75 percent or more the number of iterations of wall shapes, and therefore the tunnel run-time overhead attributable to the streamlining process, required to reach satisfactory streamlines. The procedures of the Strategy are embodied in the FORTRAN subroutine WAS (standing for Wall Adjustment Strategy) which is written in general form. The essentials of the test section hardware, followed by the underlying aerodynamic theory which forms the basis of the Strategy, are briefly described. The subroutine is then presented as the Appendix, broken down into segments with descriptions of the numerical operations underway in each, with definitions of variables

Studies of self streamlining wind tunnel real and imaginary flows

Testing in the low speed flexible walled tunnel in an effort to explain the reasons for data discrepancies at high angles of attack are presented. Automated transonic test sections were developed. The flexible walled tunnel was used in a new operating mode to a generated curved flow around the airfoil, allowing the extraction of purely rotary derivatives. Some straight wall, low speed pressure data, for wall and model, which is used for checking interference correction methods were reported. Computer software which includes an old streamlining algorithm and a prediction algorithm was examined

An Examination of US Consumer Pet and Veterinary Expenditures, 1980-1999

The veterinary medical profession touches nearly everyone's life, either directly or indirectly. An estimated 58.3% of US households own pets (AVMA, 2002), and most people consume livestock products in the form of meat, dairy products, wool, or leather. The health and well being of all these animals depend heavily on relationships with veterinarians. Veterinarians also contribute to public health through the FDA, CDC, USDA, and numerous other government agencies at the federal, state, and local levels. Issues of primary concern include food safety, biosecurity, and the numerous emerging (and re-emerging) infectious diseases that are zoonotic in nature. Finally, veterinarians have an additional impact through their research contributions. Virtually all of the laboratory animals used in research are raised, housed, and managed under the care of veterinarians, and veterinary researchers regularly provide valuable contributions to the knowledge base in the biomedical sciences. This study was designed to assess the general trends in pet and veterinary expenditures as well as factors associated with pet ownership and expenditures on veterinary medical services. Providing such key information on the sector of greatest economic importance will enhance the probability of sustained economic viability in the veterinary medical profession as a whole.Health Economics and Policy,

Streamlining the walls of an empty two-dimensional flexible-walled test section

The techniques used to find aerodynamically straight wall contours in a test section of a transonic wind tunnel are discussed. The walls were defined as aerodynamically straight up to Mach 0.9