On the growth rate of 1324-avoiding permutations

We give an improved algorithm for counting the number of $1324$-avoiding permutations, resulting in 5 further terms of the generating function. We analyse the known coefficients and find compelling evidence that unlike other classical length-4 pattern-avoiding permutations, the generating function in this case does not have an algebraic singularity. Rather, the number of 1324-avoiding permutations of length $n$ behaves as $$B\cdot \mu^n \cdot \mu_1^{n^{\sigma}} \cdot n^g.$$ We estimate $\mu=11.60 \pm 0.01,$ $\sigma=1/2,$ $\mu_1 = 0.0398 \pm 0.0010,$ $g = -1.1 \pm 0.2$ and $B =9.5 \pm 1.0.$Comment: 20 pages, 10 figure

Method for detecting pollutants

A method is described for detecting and measuring trace amounts of pollutants of the group consisting of ozone, nitrogen dioxide, and carbon monoxide in a gaseous environment. A sample organic solid material that will undergo a chemical reaction with the test pollutant is exposed to the test environment and thereafter, when heated in the temperature range of 100-200 C., undergoes chemiluminescence that is measured and recorded as a function of concentration of the test pollutant. The chemiluminescence of the solid organic material is specific to the pollutant being tested

Reply to Comments of Bassi, Ghirardi, and Tumulka on the Free Will Theorem

We show that the authors in the title have erred in claiming that our axiom FIN is false by conflating it with Bell locality. We also argue that the predictions of quantum mechanics, and in particular EPR, are fully Lorentz invariant, whereas the Free Will Theorem shows that theories with a mechanism of reduction, such as GRW, cannot be made fully invariant.Comment: We sharpen our theorem by replacing axiom FIN by a weaker axiom MIN to answer the above authors' objection

On consecutive pattern-avoiding permutations of length 4, 5 and beyond

We review and extend what is known about the generating functions for consecutive pattern-avoiding permutations of length 4, 5 and beyond, and their asymptotic behaviour. There are respectively, seven length-4 and twenty-five length-5 consecutive-Wilf classes. D-finite differential equations are known for the reciprocal of the exponential generating functions for four of the length-4 and eight of the length-5 classes. We give the solutions of some of these ODEs. An unsolved functional equation is known for one more class of length-4, length-5 and beyond. We give the solution of this functional equation, and use it to show that the solution is not D-finite. For three further length-5 c-Wilf classes we give recurrences for two and a differential-functional equation for a third. For a fourth class we find a new algebraic solution. We give a polynomial-time algorithm to generate the coefficients of the generating functions which is faster than existing algorithms, and use this to (a) calculate the asymptotics for all classes of length 4 and length 5 to significantly greater precision than previously, and (b) use these extended series to search, unsuccessfully, for D-finite solutions for the unsolved classes, leading us to conjecture that the solutions are not D-finite. We have also searched, unsuccessfully, for differentially algebraic solutions.Comment: 23 pages, 2 figures (update of references, plus web link to enumeration data). Minor update. Typos corrected. One additional referenc

High temperature, low-cycle fatigue of copper-base alloys for rocket nozzles. Part 2: Strainrange partitioning and low-cycle fatigue results at 538 deg C

Low-cycle fatigue tests of 1/2 Hard AMZIRC Copper and NARloy Z were performed in argon at 538 C to determine partitioned strain range versus life relationships. Strain-controlled low-cycle fatigue tests of a Zr-Cr-Mg copper-base alloy were also performed. Strain ranges, lower than those employed in previous tests, were imposed in order to extend the fatigue life curve out to approximately 400,000 cycles. An experimental copper alloy and an experimental silver alloy were also studied. Tensile tests were performed in air at room temperature and in argon at 538 C. Strain-controlled low-cycle fatigue tests were performed at 538 C in argon to define the fatigue life over the regime from 300 to 3,000 cycles. For the silver alloy, three additional heat treatments were introduced, and a limited evaluation of the short-term tensile and low-cycle fatigue behavior at 538 C was performed