A first step toward higher order chain rules in abelian functor calculus

One of the fundamental tools of undergraduate calculus is the chain rule. The notion of higher order directional derivatives was developed by Huang, Marcantognini, and Young, along with a corresponding higher order chain rule. When Johnson and McCarthy established abelian functor calculus, they proved a chain rule for functors that is analogous to the directional derivative chain rule when $n = 1$. In joint work with Bauer, Johnson, and Riehl, we defined an analogue of the iterated directional derivative and provided an inductive proof of the analogue to the chain rule of Huang et al. This paper consists of the initial investigation of the chain rule found in Bauer et al., which involves a concrete computation of the case when $n=2$. We describe how to obtain the second higher order directional derivative chain rule for abelian functors. This proof is fundamentally different in spirit from the proof given in Bauer et al. as it relies only on properties of cross effects and the linearization of functors

Improving Student Training Through Computer-Based Tutorials

The use of computer-based training for student employees was explored through a recent project. Tutorials were designed and created using a free software program, Wink. The tutorials covered use of electronic resources and circulation programs and were used in training newly hired and continuing student employees. Written exercises requiring use of the information covered in the tutorials were administered subsequently. The advantages of computer-based training for aspects of student library employment were contrasted with its disadvantages. Computer-based training was found to be an effective part of training student employees

Determination of the oxidative stability of perfluoropolyalkyl ethers and correlation with chemical structure

The oxidative stabilities of several perfluoropolyalkyl ethers (PFPAE) with related chemical structures were determined by thermal gravimetric analysis and correlated with their chemical structures. These results show that oxidative stability increases as the number of difluoroformal groups decreases and as trifluoromethyl substituents are added. They are also consistent with a recently proposed intramolecular disproportionation reaction mechanism involving coordination of successive ether oxygens to a Lewis acid. Since polytetrafluoroethylene contains no oxygen, it provides an indication of the upper limit to oxidative stability of PFPAE fluids. These results also show that oxidative decomposition of PFPAE fluids requires the presence of an active metal as well as air. Consequently, it may be possible to minimize decomposition and thus improve oxidative stability by passivating reactive metal surfaces

FTIR analysis of aviation fuel deposits

Five modes of operation of the Nicolet 7199 Fourier Transform Infrared Spectrophotometer have been evaluated for application in analysis of the chemical structure of accelerated storage/thermal deposits produced by jet fuels. Using primarily the absorption and emission modes, the effects of fuel type, stress temperature, stress time, type of spiking agent, spiking agent concentration, fuel flow, and post-depositional treatment on the chemical nature of fuel deposits have been determined

Trivial Meet and Join within the Lattice of Monotone Triangles

The lattice of monotone triangles $(\mathfrak{M}_n,\le)$ ordered by entry-wise comparisons is studied. Let $\tau_{\min}$ denote the unique minimal element in this lattice, and $\tau_{\max}$ the unique maximum. The number of $r$-tuples of monotone triangles $(\tau_1,\ldots,\tau_r)$ with minimal infimum $\tau_{\min}$ (maximal supremum $\tau_{\max}$, resp.) is shown to asymptotically approach $r|\mathfrak{M}_n|^{r-1}$ as $n \to \infty$. Thus, with high probability this event implies that one of the $\tau_i$ is $\tau_{\min}$ ($\tau_{\max}$, resp.). Higher-order error terms are also discussed.Comment: 15 page

Tribological characteristics of perfluoropolyether liquid lubricants under sliding conditions in high vacuum

Tribological characteristics of three PFPE's (Fomblin Z, Demnum, and Krytox) were studied under high vacuum using a four-ball apparatus with 440C steel specimens. Fomblin Z and Demnum exhibited initial scuffing-like high friction whereas Krytox did not. Steady state friction with Fomblin Z was the lowest among the three oils. Frictional values for Demnum and Krytox were almost the same. The lowest wear rate in air was provided by Krytox regardless of load, and low wear rates in vacuum at high load were achieved with Krytox and Demnum. Results are explained by reactivity and pressure-viscosity characteristics of the oils

Determination of the thermal stability of perfluoroalkylethers

The thermal decomposition temperatures of several commercial and custom synthesized perfluoroalkylether fluids were determined with a computerized tensimeter. In general, the decomposition temperatures of the commercial fluids were all similar and significantly higher than those for custom synthesized fluids. Correlation of the decomposition temperatures with the molecular structures of the primary components of the commercial fluids revealed that the stability of the fluids is not affected by intrinsic factors such as carbon chain length, branching, or cumulated difluoroformal groups. Instead, correlation with extrinsic factors revealed that the stability may be limited by the presence of small quantities of thermally unstable material and/or chlorine-containing material arising from the use of chlorine-containing solvents during synthesis. Finally, correlation of decomposition temperatures with molecular weights for Demnum and Krytox fluids supports a chain cleavage reaction mechanism for Demnum fluids and an unzipping reaction mechanism for Krytox fluids

On comparability of bigrassmannian permutations

Let Sn and Gn denote the respective sets of ordinary and bigrassmannian (BG) permutations of order n, and let (Gn,≤) denote the Bruhat ordering permutation poset. We study the restricted poset (Bn,≤), first providing a simple criterion for comparability. This criterion is used to show that that the poset is connected, to enumerate the saturated chains between elements, and to enumerate the number of maximal elements below r fixed elements. It also quickly produces formulas for β(ω) (α(ω), respectively), the number of BG permutations weakly below (weakly above, respectively) a fixed ω ∈ Bn, and is used to compute the Mo¨bius function on any interval in Bn. We then turn to a probabilistic study of β = β(ω) (α = α(ω) respectively) for the uniformly random ω ∈ Bn. We show that α and β are equidistributed, and that β is of the same order as its expectation with high probability, but fails to concentrate about its mean. This latter fact derives from the limiting distribution of β/n3. We also compute the probability that randomly chosen BG permutations form a 2- or 3-element multichain

Quantifying Spectral Features of Type Ia Supernovae

We introduce a new technique to quantify highly structured spectra for which the definition of continua or spectral features in the observed flux spectra is difficult. The method employs wavelet transformation which allows the decomposition of the observed spectra into different scales. A procedure is formulated to define the strength of spectral features so that the measured spectral indices are independent of the flux levels and are insensitive to the definition of continuum and also to reddening. This technique is applied to Type Ia supernovae spectra, where correlations are revealed between the luminosity and spectral features. The current technique may allow for luminosity corrections based on spectral features in the use of Type Ia supernovae as cosmological probe.Comment: 35 pages, 15 figure