New Lower Bounds for van der Waerden Numbers Using Distributed Computing

This paper provides new lower bounds for van der Waerden numbers. The number $W(k,r)$ is defined to be the smallest integer $n$ for which any $r$-coloring of the integers $0 \ldots, n-1$ admits monochromatic arithmetic progression of length $k$; its existence is implied by van der Waerden's Theorem. We exhibit $r$-colorings of $0\ldots n-1$ that do not contain monochromatic arithmetic progressions of length $k$ to prove that $W(k, r)>n$. These colorings are constructed using existing techniques. Rabung's method, given a prime $p$ and a primitive root $\rho$, applies a color given by the discrete logarithm base $\rho$ mod $r$ and concatenates $k-1$ copies. We also used Herwig et al's Cyclic Zipper Method, which doubles or quadruples the length of a coloring, with the faster check of Rabung and Lotts. We were able to check larger primes than previous results, employing around 2 teraflops of computing power for 12 months through distributed computing by over 500 volunteers. This allowed us to check all primes through 950 million, compared to 10 million by Rabung and Lotts. Our lower bounds appear to grow roughly exponentially in $k$. Given that these constructions produce tight lower bounds for known van der Waerden numbers, this data suggests that exact van der Waerden Numbers grow exponentially in $k$ with ratio $r$ asymptotically, which is a new conjecture, according to Graham.Comment: 8 pages, 1 figure. This version reflects new results and reader comment

How to Probe for Dynamical Structure in the Collapse of Entangled States Using Nuclear Magnetic Resonance

The spin state of two magnetically inequivalent protons in contiguous atoms of a molecule becomes entangeled by the indirect spin-spin interaction (j-coupling). The degree of entanglement oscillates at the beat frequency resulting from the splitting of a degeneracy. This beating is manifest in NMR spectroscopy as an envelope of the transverse magnetization and should be visible in the free induction decay signal. The period (approximately 1 sec) is long enough for interference between the linear dynamics and collapse of the wave-function induced by a Stern-Gerlach inhomogeneity to significantly alter the shape of that envelope. Various dynamical collapse theories can be distinguished by their observably different predictions with respect to this alteration. Adverse effects of detuning due to the Stern-Gerlach inhomogeneity can be reduced to an acceptable level by having a sufficiently thin sample or a strong rf field.Comment: 6 pages, 4 figures, PDF, submitted to PR

The Subtleties of Social Exclusion: Race, Social Class, and the Exclusion of Blacks in a Racially Mixed Neighborhood

We use interviews, content analysis, and surveys to describe how a neighborhood association in a racially mixed neighborhood in Portland, Oregon (USA) subtly excludes many blacks from being full members of the neighborhood. In contrast to explicit cases of social exclusion, this neighborhood association excludes blacks without ever referring to race. They instead justify their actions—e.g., helping close down a black social club and discouraging more affordable housing—based on such nonracial goals as increasing homeownership, minimizing crime, and maximizing “economic diversity.” We argue that without the inclusion of black residents and their neighborhood organizations (e.g., churches) in the decision-making process, mixed-race neighborhoods will continue to lose their diversity

Effect of Stator and Rotor Aspect Ratio on Transonic-Turbine Performance

The effect of stator and rotor aspect ratio on transonic-turbine performance was experimentally investigated. The stator aspect ratios covered were 1.6. 0.8, and 0.4, while the rotor aspect ratios investigated were 1.46 and 0.73. It was found that the observed variation in turbine design-point efficiency was negligible. Thus, within the range of aspect ratio investigated, these results verify for turbines operating in the transonic flow range the finding of a reference report, which showed analytically that, if blade shape and solidity are held constant, the aspect ratio may be varied over a wide range without appreciable change in turbine efficiency

Experimental Investigation of Flow in an Annular Cascade of Turbine Nozzle Blades of Constant Discharge Angle

The experimental performance of turbine nozzle blades designed for a constant discharge angle was investigated at discharge hub Mach numbers of 1.18, 1.31, and 1.41. Flow characteristics are presented in terms of energy losses, angle gradients, and secondary flow effects. Blade efficiency decreased from 0.983 to 0.978 with increasing Mach number in the range investigated while angle variations in the loss regions became very large, indicating poorer blade performance than efficiency implies